Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

Abstract

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

1. Introduction

We analyze the system of Hadamard fractional differential equations

$$\left\{\begin{array}{c}{}^{H}D_{1+}^{\alpha }\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{f}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\mathsf{\xi }\right))=0,\mathsf{\xi }\in [1,e],\hfill \\ {}^{H}D_{1+}^{\beta }\mathfrak{v}\left(\mathsf{\xi }\right)+\mathfrak{g}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\mathsf{\xi }\right))=0,\mathsf{\xi }\in [1,e],\hfill \end{array}\right.$$

(1)

supplemented with the coupled nonlocal boundary conditions

$$\left\{\begin{array}{c}{\mathfrak{u}}^{\left(i\right)}\left(1\right)=0,i=0,\dots ,n-2,{\mathfrak{v}}^{\left(i\right)}\left(1\right)=0,i=0,\dots ,m-2,\hfill \\ {\displaystyle {}^{H}D_{1+}^{\varsigma }\mathfrak{u}\left(e\right)=\sum _{i=1}^a{\int }_1^e{}^{H}D_{1+}^{{\varrho }_i}\mathfrak{u}\left(\nu \right)d{H}_i\left(\nu \right)+\sum _{i=1}^b{\int }_1^e{}^{H}D_{1+}^{{\sigma }_i}\mathfrak{v}\left(\nu \right)d{K}_i\left(\nu \right),}\hfill \\ {\displaystyle {}^{H}D_{1+}^{\vartheta }\mathfrak{v}\left(e\right)=\sum _{i=1}^c{\int }_1^e{}^{H}D_{1+}^{{\eta }_i}\mathfrak{u}\left(\nu \right)d{P}_i\left(\nu \right)+\sum _{i=1}^d{\int }_1^e{}^{H}D_{1+}^{{\theta }_i}\mathfrak{v}\left(\nu \right)d{Q}_i\left(\nu \right),}\hfill \end{array}\right.$$

(2)

where $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$, $\beta \in (m-1,m]$, $m\in \mathbb{N}$, $m\ge 3$; $a,b,c,d\in \mathbb{N}$; $\varsigma \ge 1$, $0\le {\varrho }_i\le \varsigma <\alpha -1$, $i=1,\dots ,a$, $0\le {\eta }_k\le \varsigma$, $k=1,\dots ,c$, $\vartheta \ge 1$, $0\le {\sigma }_j\le \vartheta <\beta -1$, $j=1,\dots ,b$, $0\le {\theta }_{\iota }\le \vartheta$, $\iota =1,\dots ,d$; ${\gamma }_i,{\delta }_i>0$, $i=1,2$; ${}^{H}D_{1+}^{\kappa }$ (for $\kappa =\alpha ,\varsigma ,\beta ,\vartheta ,{\varrho }_i,{\sigma }_j,{\eta }_k,{\theta }_{\iota },$ $i=1,\dots ,a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$) is the Hadamard fractional derivative of order $\kappa$; and ${}^{H}I_{1+}^{\kappa }$ (for $\kappa ={\gamma }_i,{\delta }_i,i=1,2$) is the Hadamard fractional integral of order $\kappa$. The integrals from the boundary conditions (2) are Riemann–Stieltjes integrals with $H_i,K_j,P_k,Q_{\iota }$, $i=1,\dots ,a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$, bounded variation functions. The functions $\mathfrak{f},\mathfrak{g}:[1,e]\times {\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ are given ones, (${\mathbb{R}}_{+}=[0,\infty )$), and $\mathfrak{u},\mathfrak{v}:[1,e]\to \mathbb{R}$ are the unknown functions of Problem (1),(2).

We give various assumptions on the functions $\mathfrak{f}$ and $\mathfrak{g}$ such that Problem (1),(2) has at least one positive solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right),\mathsf{\xi }\in [1,e]$. For the proof of our main results, we use several fixed-point theorems, such as the Banach contraction mapping principle, the Guo–Krasnosel’skii fixed-point theorem, the Schauder fixed-point theorem, and the Leggett–Williams fixed-point theorem. In what follows, we present some papers that are connected to our Problem (1),(2). In [1], the authors examined the system of nonlinear Hadamard fractional differential equations on the infinite interval $[1,\infty )$, namely,

$$\left\{\begin{array}{c}{}^{H}D_{1+}^{\alpha }\mathrm{x}\left(\nu \right)+\mathfrak{a}\left(\nu \right)\mathrm{f}(\mathrm{x}\left(\nu \right),\mathrm{y}\left(\nu \right))=0,\nu \in (1,\infty ),\hfill \\ {}^{H}D_{1+}^{\beta }\mathrm{y}\left(\nu \right)+\mathfrak{b}\left(\nu \right)\mathrm{g}(\mathrm{x}\left(\nu \right),\mathrm{y}\left(\nu \right))=0,\nu \in (1,\infty ),\hfill \end{array}\right.$$

(3)

subject to the nonlocal boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \mathrm{x}\left(1\right)={\mathrm{x}}^{\prime }\left(1\right)=\cdots ={\mathrm{x}}^{(n-2)}\left(1\right)=0,{}^{H}D_{1+}^{\alpha -1}\mathrm{x}(\infty )={\int }_1^{\infty }\mathrm{x}\left(\rho \right)d{\mathfrak{H}}_1\left(\rho \right)+{\int }_1^{\infty }\mathrm{y}\left(\rho \right)d{\mathfrak{H}}_2\left(\rho \right),}\hfill \\ {\displaystyle \mathrm{y}\left(1\right)={\mathrm{y}}^{\prime }\left(1\right)=\cdots ={\mathrm{y}}^{(m-2)}\left(1\right)=0,{}^{H}D_{1+}^{\beta -1}\mathrm{y}(\infty )={\int }_1^{\infty }\mathrm{x}\left(\rho \right)d{\mathfrak{K}}_1\left(\rho \right)+{\int }_1^{\infty }\mathrm{y}\left(\rho \right)d{\mathfrak{K}}_2\left(\rho \right),}\hfill \end{array}\right.$$

(4)

where $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 2$; $\beta \in (m-1,m]$, $m\in \mathbb{N}$, $m\ge 2$; the functions $\mathrm{f},\mathrm{g}$ are nonnegative and bounded; and the functions ${\mathfrak{H}}_1,{\mathfrak{H}}_2,{\mathfrak{K}}_1,{\mathfrak{K}}_2:[1,\infty )\to \mathbb{R}$ have bounded variations. Using the Leggett–Williams fixed-point theorem and the Guo–Krasnosel’skii fixed-point theorem, the authors proved the existence of positive solutions to Problem (3),(4). In paper [2], the authors investigated Problem (3),(4) in the case in which the functions $\mathrm{f},\mathrm{g}$ are unbounded, and in the main results, they relied on the Banach contraction mapping principle, the Schauder fixed-point theorem, and the Avery–Peterson fixed-point theorem. In [3], the authors analyzed the existence and uniqueness of solutions to the system of Hilfer–Hadamard fractional differential equations

$$\left\{\begin{array}{c}{}^{\mathit{HH}}D_1^{\alpha ,\beta }\mathfrak{u}\left(\mathsf{\xi }\right)=\mathfrak{f}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [1,T],\hfill \\ {}^{\mathit{HH}}D_1^{\gamma ,\delta }\mathfrak{v}\left(\mathsf{\xi }\right)=\mathfrak{g}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [1,T],\hfill \end{array}\right.$$

with the nonlocal coupled boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \mathfrak{u}\left(1\right)=0,{}^{H}D_1^{\varsigma }\mathfrak{u}\left(T\right)=\sum _{i=1}^m{\int }_1^T{}^{H}D_1^{{\varrho }_i}\mathfrak{u}\left(\rho \right)d{H}_i\left(\rho \right)+\sum _{i=1}^n{\int }_1^T{}^{H}D_1^{{\sigma }_i}\mathfrak{v}\left(\rho \right)d{K}_i\left(\rho \right),}\hfill \\ {\displaystyle \mathfrak{v}\left(1\right)=0,{}^{H}D_1^{\vartheta }\mathfrak{v}\left(T\right)=\sum _{i=1}^p{\int }_1^T{}^{H}D_1^{{\eta }_i}\mathfrak{u}\left(\rho \right)d{P}_i\left(\rho \right)+\sum _{i=1}^q{\int }_1^T{}^{H}D_1^{{\theta }_i}\mathfrak{v}\left(\rho \right)d{Q}_i\left(\rho \right),}\hfill \end{array}\right.$$

(5)

where $T>1$; $\beta ,\delta \in [0,1]$; $\alpha ,\gamma \in (1,2]$; $m,n,p,q\in \mathbb{N}$; $\varsigma ,\vartheta ,{\varrho }_i,{\sigma }_i,{\eta }_i,{\theta }_i\in [0,1]$; ${}^{\mathit{HH}}D_1^{{\kappa }_1,{\kappa }_2}$ denotes the Hilfer–Hadamard fractional derivative of order ${\kappa }_1$ and type ${\kappa }_2$ (for ${\kappa }_1=\alpha ,\gamma$ and ${\kappa }_2=\beta ,\delta$); the continuous functions $\mathfrak{f}$ and $\mathfrak{g}$ are defined on $[1,T]\times {\mathbb{R}}^2$; and the integrals from the boundary conditions (5) are Riemann–Stieltjes integrals with $H_i,K_j,P_k,Q_{\iota }$, $i=1,\dots ,m,$ $j=1,\dots ,n$, $k=1,\dots ,p$, and $\iota =1,\dots ,q$ functions of bounded variation. In the proof of the main results, they used the Krasnosel’skii fixed-point theorem applied to the sum of two operators, the Banach contraction mapping principle, the Leray–Schauder nonlinear alternative, and the Schaefer fixed-point theorem. In the paper [4], the authors studied the system of random sequential fractional differential equations with integral boundary conditions

$$\left\{\begin{array}{c}{\mathrm{D}}^{\alpha }({\mathrm{D}}^{\beta }+{\lambda }_1)\mathrm{x}(s,\omega )=\mathfrak{f}(s,\mathrm{x}(s,\omega ),\mathrm{y}(s,\omega ),\omega ),s\in [1,e],\hfill \\ {\mathrm{D}}^{\gamma }({\mathrm{D}}^{\sigma }+{\lambda }_2)\mathrm{y}(s,\omega )=\mathfrak{g}(s,\mathrm{x}(s,\omega ),\mathrm{y}(s,\omega ),\omega ),s\in [1,e],\hfill \\ {\displaystyle \sum _{i=1}^m{\theta }_i{\mathrm{I}}^{{\mu }_i}\mathrm{x}({\eta }_i,\omega )=\sum _{j=1}^n{\varphi }_j{\mathrm{I}}^{{\gamma }_j}\mathrm{x}({\mathsf{\xi }}_j,\omega ),\sum _{k=1}^p{ϵ}_k{\mathrm{I}}^{{\varsigma }_k}\mathrm{x}({\psi }_k,\omega )=\sum _{l=1}^q{\nu }_l{\mathrm{I}}^{{\tau }_l}\mathrm{x}({\phi }_l,\omega ),}\hfill \\ {\displaystyle \sum _{i=1}^m{\overline{\theta }}_i{\mathrm{I}}^{{\overline{\mu }}_i}\mathrm{y}({\overline{\eta }}_i,\omega )=\sum _{j=1}^n{\overline{\varphi }}_j{\mathrm{I}}^{{\overline{\gamma }}_j}\mathrm{y}({\overline{\mathsf{\xi }}}_j,\omega ),\sum _{k=1}^p{\overline{ϵ}}_k{\mathrm{I}}^{{\overline{\varsigma }}_k}\mathrm{y}({\overline{\psi }}_k,\omega )=\sum _{l=1}^q{\overline{\nu }}_l{\mathrm{I}}^{{\overline{\tau }}_l}\mathrm{y}({\overline{\phi }}_l,\omega ),}\hfill \end{array}\right.$$

(6)

where $\alpha ,\beta ,\gamma ,\sigma \in (0,1)$, $\alpha +\beta \in (1,2)$; $\gamma +\sigma \in (1,2)$; the constants ${\lambda }_1,{\lambda }_2$ are given ones; the constants ${\eta }_i,{\overline{\eta }}_i,{\mathsf{\xi }}_j,{\overline{\mathsf{\xi }}}_j,{\psi }_k,{\overline{\psi }}_k,{\phi }_l,{\overline{\phi }}_l\in (1,e)$; ${\theta }_i,{\overline{\theta }}_i,{\varphi }_j,{\overline{\varphi }}_j,{ϵ}_k,{\overline{ϵ}}_k,{\nu }_l,{\overline{\nu }}_l\in \mathbb{R}$ for $i=1,\dots ,m$, $j=1,\dots ,n$, $k=1,\dots ,p$, $l=1,\dots ,q$; $\mathfrak{f},\mathfrak{g}:[1,e]\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathsf{\Omega }\to {\mathbb{R}}^m$; $(\mathsf{\Omega },\mathcal{A})$ is a measurable space; and ${\mathrm{D}}^q$ and ${\mathrm{I}}^q$ are the Hadamard–Caputo fractional derivative and Hadamard fractional integral, respectively. Based on the random fixed-point principles of Perov and Schaefer and a vector approach that uses matrices convergent to zero, they authors showed the existence and uniqueness of solutions for Problem (6). In paper [5], the authors examined the Hadamard fractional boundary value problem at resonance

$$\left\{\begin{array}{c}{}^{H}D_{1+}^{\gamma }\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{f}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right))=0,\mathsf{\xi }\in (1,e),\hfill \\ \mathfrak{u}\left(1\right)=0,\mathfrak{u}\left(e\right)={\int }_1^e\mathfrak{u}\left(\nu \right)dA\left(\nu \right),\hfill \end{array}\right.$$

(7)

where $\gamma \in (1,2)$ and the function $\mathfrak{f}:[1,e]\times {\mathbb{R}}^2\to \mathbb{R}$ satisfies Caratheodory conditions. By using the coincidence degree theory of Mawhin and the method of constructing appropriate operators, the authors proved the existence of solutions to Problem (7).

The Hadamard fractional calculus has applications in various fields, such as rheology and probability (see papers [6,7] and their references). For example, the generalizations of the Lomnitz logarithmic creep law describes geophysical phenomena to fluid-like materials including igneous rocks. For additional recent studies on Hadamard derivatives, integrals, and fractional differential equations and systems, we recommend the monograph [8] and its references, as well as papers [9,10,11,12,13,14,15,16,17,18,19,20,21,22].

The novelty of our Problem (1),(2) consists of the consideration of the system of Hadamard fractional differential equations with derivatives of diverse orders and the presence of fractional integral terms in the nonlinearities of the system. In addition, the last two conditions from (2) are very general, encompassing cases of classical integral boundary conditions, multi-point boundary conditions (when the functions $H_i,K_j,P_k,Q_{\iota }$, for $i=1,\dots a$, $j=1,\dots ,b$, $k=1,\dots ,c$, $\iota =1,\dots ,d$ are step functions), and combinations of them. A very important remark is the fact that, in the penultimate condition of (2), the Hadamard fractional derivative of order $\varsigma$ of function $\mathfrak{u}$ at point e is dependent on the fractional derivatives of various orders of both $\mathfrak{u}$ and $\mathfrak{v}$ over the entire interval $[1,e]$. And, in the last condition of (2), the Hadamard fractional derivative of order $\vartheta$ of function $\mathfrak{v}$ at point e depends on the fractional derivatives of varied orders of both the functions $\mathfrak{u}$ and $\mathfrak{v}$ on the entire interval $[1,e]$. We also note that the Hadamard fractional integrals that appear in the boundary conditions of some fractional boundary value problems in the literature can be written as Riemann–Stieltjes integrals, similar to those in our conditions (2) (see [1]). All these remarks lead to the conclusion that Problem (1),(2) are novel in the context of the systems of coupled Hadamard boundary value problems.

This paper is organized as follows. In Section 2, we examine the linear boundary value problem associated to Problem (1),(2) and present the corresponding Green functions and some of their properties. Section 3 outlines the main theorems regarding the existence of positive solutions to our problem. Section 4 provides examples that illustrate the main results, and finally, Section 5 offers the conclusions of the paper.

2. Preliminary Results

We consider the linear problem associated to our Problem (1),(2), namely, the system of Hadamard fractional differential equations

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

(8)

with the boundary conditions (2), where $\mathfrak{h},\mathfrak{k}\in C\left(\right[1,e\left]\right)$.

We denote

$$\begin{array}{c}{\displaystyle {\mathsf{\Xi }}_1=\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\varsigma )}-\sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\nu \right),}\hfill \\ {\displaystyle {\mathsf{\Xi }}_2=\sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right),}\hfill \\ {\displaystyle {\mathsf{\Xi }}_3=\sum _{i=1}^c\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\nu \right),}\hfill \\ {\displaystyle {\mathsf{\Xi }}_4=\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}-\sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right),}\hfill \\ \Delta ={\mathsf{\Xi }}_1{\mathsf{\Xi }}_4-{\mathsf{\Xi }}_2{\mathsf{\Xi }}_3.\hfill \end{array}$$

(9)

Lemma 1. 

If $\Delta \ne 0$, then the solution of Problem (8),(2) is given by

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)=-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }({\mathsf{\Xi }}_{4}L+{\mathsf{\Xi }}_{2}M),\mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)=-\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\beta -1}}{\Delta }({\mathsf{\Xi }}_{3}L+{\mathsf{\Xi }}_{1}M),\mathsf{\xi }\in [1,e],}\hfill \end{array}$$

(10)

where

$$\begin{array}{c}{\displaystyle L=\frac{1}{\mathsf{\Gamma }(\alpha -\varsigma )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{H}_i\left(\nu \right)}\hfill \\ {\displaystyle -\sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\sigma }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{K}_i\left(\nu \right),}\hfill \\ {\displaystyle M=\frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\eta }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{P}_i\left(\nu \right)}\hfill \\ {\displaystyle -\sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\theta }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{Q}_i\left(\nu \right).}\hfill \end{array}$$

(11)

Proof. 

To the equations of system (8), we apply the integral operators ${}^{H}I_{1+}^{\alpha }$ and ${}^{H}I_{1+}^{\beta }$, respectively. Then, the solutions of System (8) are given by

$$\begin{array}{c}\mathfrak{u}\left(\mathsf{\xi }\right)={-}^H{I}_{1+}^{\alpha }\mathfrak{h}\left(\mathsf{\xi }\right)+a_1{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}+a_2{\left(\ln \mathsf{\xi }\right)}^{\alpha -2}+\cdots +a_n{\left(\ln \mathsf{\xi }\right)}^{\alpha -n}\hfill \\ {\displaystyle =-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }ds+a_1{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}+a_2{\left(\ln \mathsf{\xi }\right)}^{\alpha -2}+\cdots +a_n{\left(\ln \mathsf{\xi }\right)}^{\alpha -n},}\hfill \\ \mathfrak{v}\left(\mathsf{\xi }\right)={-}^H{I}_{1+}^{\beta }\mathfrak{k}\left(\mathsf{\xi }\right)+b_1{\left(\ln \mathsf{\xi }\right)}^{\beta -1}+b_2{\left(\ln \mathsf{\xi }\right)}^{\beta -2}+\cdots +b_m{\left(\ln \mathsf{\xi }\right)}^{\beta -m}\hfill \\ {\displaystyle =-\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu +b_1{\left(\ln \mathsf{\xi }\right)}^{\beta -1}+b_2{\left(\ln \mathsf{\xi }\right)}^{\beta -2}+\cdots +b_m{\left(\ln \mathsf{\xi }\right)}^{\beta -m},}\hfill \end{array}$$

(12)

for all $\mathsf{\xi }\in [1,e]$, with $a_i,b_j\in \mathbb{R}$, $i=1,\dots ,n$, $j=1,\dots ,m$. Because ${\mathfrak{u}}^{\left(i\right)}\left(1\right)=0$ for all $i=0,\dots ,n-2$ and ${\mathfrak{v}}^{\left(j\right)}\left(1\right)=0$ for all $j=0,\dots ,m-2$, we obtain $a_i=0$ for all $i=2,\dots ,n$ and $b_j=0$ for all $j=2,\dots ,m$. Therefore, by (12), we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)=-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +a_1{\left(\ln \mathsf{\xi }\right)}^{\alpha -1},\mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)=-\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu +b_1{\left(\ln \mathsf{\xi }\right)}^{\beta -1},\mathsf{\xi }\in [1,e].}\hfill \end{array}$$

(13)

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

$$\begin{array}{c}{}^{H}D_{1+}^{\kappa }\mathfrak{u}\left(\mathsf{\xi }\right)={-}^H{D}_{1+}^{\kappa }{}^{H}I_{1+}^{\alpha }\mathfrak{h}\left(\mathsf{\xi }\right)+a_1{}^{H}D_{1+}^{\kappa }\left({\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\right)\hfill \\ {\displaystyle ={-}^H{I}_{1+}^{\alpha -\kappa }\mathfrak{h}\left(\mathsf{\xi }\right)+a_1\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\kappa )}{\left(\ln \mathsf{\xi }\right)}^{\alpha -\kappa -1}}\hfill \\ {\displaystyle =-\frac{1}{\mathsf{\Gamma }(\alpha -\kappa )}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -\kappa -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +a_1\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\kappa )}{\left(\ln \mathsf{\xi }\right)}^{\alpha -\kappa -1},}\hfill \end{array}$$

(14)

and for $\tilde{\kappa }=\vartheta$, $\tilde{\kappa }={\sigma }_i,i=1,\dots ,b$, and $\tilde{\kappa }={\theta }_j,j=1,\dots ,d$, we find

$$\begin{array}{c}{}^{H}D_{1+}^{\tilde{\kappa }}\mathfrak{v}\left(\mathsf{\xi }\right)={-}^H{D}_{1+}^{\tilde{\kappa }}{}^{H}I_{1+}^{\beta }\mathfrak{k}\left(\mathsf{\xi }\right)+b_1{}^{H}D_{1+}^{\tilde{\kappa }}\left({\left(\ln \mathsf{\xi }\right)}^{\beta -1}\right)\hfill \\ {\displaystyle ={-}^H{I}_{1+}^{\beta -\tilde{\kappa }}\mathfrak{k}\left(\mathsf{\xi }\right)+b_1\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\tilde{\kappa })}{\left(\ln \mathsf{\xi }\right)}^{\beta -\tilde{\kappa }-1}}\hfill \\ {\displaystyle =-\frac{1}{\mathsf{\Gamma }(\beta -\tilde{\kappa })}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -\tilde{\kappa }-1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu +b_1\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\tilde{\kappa })}{\left(\ln \mathsf{\xi }\right)}^{\beta -\tilde{\kappa }-1}.}\hfill \end{array}$$

(15)

By applying the last two conditions of (2) to the functions u and v from (13) and by using Relations (14) and (15), we deduce

$$\begin{array}{c}{\displaystyle -\frac{1}{\mathsf{\Gamma }(\alpha -\varsigma )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +a_1\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\varsigma )}}\hfill \\ {\displaystyle =\sum _{i=1}^a{\int }_1^e\left[{\displaystyle -\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho +a_1\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \nu \right)}^{\alpha -{\varrho }_i-1}}\right]d{H}_i\left(\nu \right)}\hfill \\ {\displaystyle +\sum _{i=1}^b{\int }_1^e\left[{\displaystyle -\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^{\nu }{\left({\displaystyle \ln \frac{s}{\rho }}\right)}^{\beta -{\sigma }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho +b_1\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}}\right]d{K}_i\left(\nu \right),}\hfill \\ {\displaystyle -\frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu +b_1\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}}\hfill \\ {\displaystyle =\sum _{i=1}^c{\int }_1^e\left[{\displaystyle -\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\eta }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho +a_1\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}}\right]d{P}_i\left(\nu \right)}\hfill \\ {\displaystyle +\sum _{i=1}^d{\int }_1^e\left[{\displaystyle -\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^s{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\theta }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho +b_1\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}}\right]d{Q}_i\left(\nu \right).}\hfill \end{array}$$

(16)

The system (16) can be written equivalently as

$$\left\{\begin{array}{c}{\mathsf{\Xi }}_1{a}_1-{\mathsf{\Xi }}_2{b}_1=L,\hfill \\ -{\mathsf{\Xi }}_3{a}_1+{\mathsf{\Xi }}_4{b}_1=M,\hfill \end{array}\right.$$

(17)

where ${\mathsf{\Xi }}_i,i=1,\dots ,4$, is given by (9) and $L,M$ are given by (11). System (17) in the unknowns $a_1$ and $b_1$ has the determinant $\Delta ={\mathsf{\Xi }}_1{\mathsf{\Xi }}_4-{\mathsf{\Xi }}_2{\mathsf{\Xi }}_3$, (given by (9)), which is different from zero. Then, the solution of system (17) is unique, given by

$$\left\{\begin{array}{c}{\displaystyle a_1=\frac{1}{\Delta }({\mathsf{\Xi }}_{4}L+{\mathsf{\Xi }}_{2}M),}\hfill \\ {\displaystyle b_1=\frac{1}{\Delta }({\mathsf{\Xi }}_{3}L+{\mathsf{\Xi }}_{1}M).}\hfill \end{array}\right.$$

(18)

By replacing the formulas for $a_1$ and $b_1$ (from (18)) in (13), we find the solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right)$, $\mathsf{\xi }\in [1,e]$ of Problem (8),(2), given by (10). □

Lemma 2. 

If $\Delta \ne 0$, then the solution of Problem (8),(2) (given by (10)) can be expressed as

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\mathsf{\xi }\in [1,e],}\hfill \end{array}$$

(19)

where the Green functions ${\mathcal{G}}_i,i=1,\dots ,4$, are given by

$$\begin{array}{c}{\displaystyle {\mathcal{G}}_1(\mathsf{\xi },\nu )=g_1(\mathsf{\xi },\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right],}\hfill \\ {\displaystyle {\mathcal{G}}_2(\mathsf{\xi },\nu )=\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right],}\hfill \\ {\displaystyle {\mathcal{G}}_3(\mathsf{\xi },\nu )=\frac{{\left(\ln \mathsf{\xi }\right)}^{\beta -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_3\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)+{\mathsf{\Xi }}_1\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right],}\hfill \\ {\displaystyle {\mathcal{G}}_4(\mathsf{\xi },\nu )=g_2(\mathsf{\xi },\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{\beta -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_3\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_1\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right],}\hfill \end{array}$$

(20)

for all $\mathsf{\xi },\nu \in [1,e]$, with

$$\begin{gathered} {\displaystyle g_1(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \\ {\displaystyle g_2(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \\ {\displaystyle g_{1i}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -{\varrho }_i-1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \\ {\displaystyle g_{2j}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_j)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\eta }_j-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -{\eta }_j-1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\alpha -{\eta }_j-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \\ {\displaystyle g_{3k}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_k)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\beta -{\sigma }_k-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -{\sigma }_k-1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\beta -{\sigma }_k-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \\ {\displaystyle g_{4\iota }(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_{\iota })}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{\beta -{\theta }_{\iota }-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\beta -{\theta }_{\iota }-1},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{\beta -{\theta }_{\iota }-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.} \end{gathered}$$

(21)

for all $\mathsf{\xi },\nu \in [1,e]$, $i=1,\dots ,a$, $j=1,\dots ,c$, $k=1,\dots ,b$, $\iota =1,\dots ,d$.

Proof. 

By Lemma 1 (Formula (10)), we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)=\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu -\frac{\Delta }{\mathsf{\Gamma }\left(\alpha \right)\Delta }{\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }({\mathsf{\Xi }}_{4}L+{\mathsf{\Xi }}_{2}M)}\hfill \\ =\underset{L_0}{\underbrace{{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu -\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }}}\hfill \\ {\displaystyle -\frac{\Delta }{\mathsf{\Gamma }\left(\alpha \right)\Delta }{\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }({\mathsf{\Xi }}_{4}L+{\mathsf{\Xi }}_{2}M)}\hfill \\ {\displaystyle =L_0-\frac{1}{\Delta \mathsf{\Gamma }\left(\alpha \right)}({\mathsf{\Xi }}_1{\mathsf{\Xi }}_4-{\mathsf{\Xi }}_2{\mathsf{\Xi }}_3){\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }({\mathsf{\Xi }}_{4}L+{\mathsf{\Xi }}_{2}M),\forall \mathsf{\xi }\in [1,e].}\hfill \end{array}$$

By replacing the formulas for $L,M$, and ${\mathsf{\Xi }}_i,i=1,\dots ,4$, we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)=L_0+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left\{-\left[\left({\displaystyle \frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\varsigma )}-\sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\nu \right)}\right)\right.\right.}\hfill \\ \times \left({\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}-\sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\hfill \\ -\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\hfill \\ {\displaystyle \times \left.\left({\displaystyle \sum _{i=1}^c\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\nu \right)}\right)\right]\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ +\left({\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}-\sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\hfill \\ \times \left[{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -\varsigma )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu -\sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{H}_i\left(\nu \right)}\right.\hfill \\ -\left.{\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\sigma }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{K}_i\left(\nu \right)}\right]\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }ds}\right.\hfill \\ {\displaystyle -\sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\eta }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{P}_i\left(\nu \right)}\hfill \\ -\left.\left.{\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\theta }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{Q}_i\left(\nu \right)}\right]\right\}\hfill \\ {\displaystyle =L_0+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left\{{\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\alpha -\varsigma )\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right.}\hfill \\ {\displaystyle +\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\varsigma )}\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\nu \right)}\right)\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ -\left({\displaystyle \sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\hfill \\ {\displaystyle \times \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)}\hfill \\ {\displaystyle \times \left({\displaystyle \sum _{i=1}^c\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\nu \right)}\right)\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )\mathsf{\Gamma }(\alpha -\varsigma )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{H}_i\left(\nu \right)}\right)}\hfill \\ {\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\sigma }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{K}_i\left(\nu \right)}\right)}\hfill \\ -\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\left({\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -\varsigma )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{H}_i\left(\nu \right)}\right)\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\sigma }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{K}_i\left(\nu \right)}\right)\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left({\displaystyle \frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ -\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\eta }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{P}_i\left(\nu \right)}\right)\hfill \\ -\left.\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\beta -{\theta }_i-1}\frac{\mathfrak{k}\left(\rho \right)}{\rho }d\rho }\right)d{Q}_i\left(\nu \right)}\right)\right\}.\hfill \end{array}$$

(22)

For the above double integrals, we change the order of the integrals by using the formulas

$$\begin{array}{c}{\displaystyle {\int }_1^e\left({\displaystyle {\int }_1^{\nu }{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\right)d{H}_i\left(\nu \right)={\int }_1^e\left({\displaystyle {\int }_{\rho }^e{\left({\displaystyle \ln \frac{\nu }{\rho }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\nu \right)}\right)\frac{\mathfrak{h}\left(\rho \right)}{\rho }d\rho }\hfill \\ {\displaystyle ={\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu ,i=1,\dots ,a,}\hfill \end{array}$$

and similar ones for the double integrals with functions $K_j$, $j=1,\dots ,b$, $P_k$, $k=1,\dots ,c$, and $Q_{\iota }$, $\iota =1,\dots ,d$. In addition, by changing the notation of the variable (from s to $\rho$) in the simple integrals and by grouping the terms according to the functions $\mathfrak{h}$ and $\mathfrak{k}$ in (22), we find

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)=L_0+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }}\hfill \\ \times \left\{{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right.\hfill \\ -\left({\displaystyle \sum _{i=1}^a\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle \times \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right)}\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ -\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\rho \right)}\right)\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^c\frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\nu \right)}\right)\hfill \\ {\displaystyle \times \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)}\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\int }_1^e{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ -\left.\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e\left({\displaystyle {\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\right\}\hfill \\ {\displaystyle =L_0+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left\{{\displaystyle {\int }_1^e\left[{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right){\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}}\right.}\right.}\hfill \\ -\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right){\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\nu \right)}\right)\left({\displaystyle \sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \nu \right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\nu \right)}\right){\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\hfill \\ {\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)}\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle -\left.\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\rho \right)}\right)\right]\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +{\int }_1^e\left[{\displaystyle -\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)}\right.}\hfill \\ +\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\frac{1}{\mathsf{\Gamma }(\beta -\vartheta )}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}}\hfill \\ -\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\hfill \\ -\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{K}_i\left(\rho \right)}\right)\hfill \\ +\left.{\displaystyle \left.\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\left({\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{Q}_i\left(\rho \right)}\right)\right]\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right\}.\hfill \end{array}$$

The last two terms were added by us, which together give zero. Now, by grouping the terms, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)={\int }_1^e{g}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }}\hfill \\ \times \left\{{\displaystyle {\int }_1^e\left\{{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\sum _{i=1}^a\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}d{H}_i\left(\rho \right)}\right.}\right.}\right.\hfill \\ -\left.{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right]-\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle \times \sum _{i=1}^a\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}d{H}_i\left(\rho \right)-\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right]}\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle \left.{\displaystyle \times \sum _{i=1}^c\left[{\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\eta }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}d{P}_i\left(\rho \right)-\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\rho \right)}\right]}\right\}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +{\int }_1^e\left\{{\displaystyle \frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -\vartheta )}\sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{K}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right]}\right.}\hfill \\ -\left({\displaystyle \sum _{i=1}^d\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\theta }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right)\hfill \\ {\displaystyle \times \sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{K}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right]}\hfill \\ +\left({\displaystyle \sum _{i=1}^b\frac{\mathsf{\Gamma }\left(\beta \right)}{\mathsf{\Gamma }(\beta -{\sigma }_i)}{\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right)\hfill \\ \times \left.{\displaystyle \left.{\displaystyle \sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{Q}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right]}\right\}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right\}\hfill \\ {\displaystyle ={\int }_1^e{g}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }}\hfill \\ \times \left\{{\displaystyle {\int }_1^e\left\{{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^a\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}d{H}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\varrho }_i-1}d{H}_i\left(\rho \right)}\right]}\right.}\right.\hfill \\ {\displaystyle \left.{\displaystyle +{\mathsf{\Xi }}_2\sum _{i=1}^c\frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\alpha -{\eta }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}d{P}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\alpha -{\eta }_i-1}d{P}_i\left(\rho \right)}\right]}\right\}\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle +{\int }_1^e\left\{{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^b\frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\beta -{\sigma }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{K}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\sigma }_i-1}d{K}_i\left(\rho \right)}\right]}\right.}\hfill \\ \left.{\displaystyle \left.{\displaystyle +{\mathsf{\Xi }}_2\sum _{i=1}^d\frac{1}{\mathsf{\Gamma }(\beta -{\theta }_i)}\left[{\displaystyle {\int }_1^e{\left(\ln \rho \right)}^{\beta -{\theta }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\beta -\vartheta -1}d{Q}_i\left(\rho \right)-{\int }_{\nu }^e{\left({\displaystyle \ln \frac{\rho }{\nu }}\right)}^{\beta -{\theta }_i-1}d{Q}_i\left(\rho \right)}\right]}\right\}\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right\}.\hfill \end{array}$$

Therefore, we deduce

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)={\int }_1^e{g}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left\{{\displaystyle {\int }_1^e\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)}\right.}\right.}\hfill \\ {\displaystyle +\left.{\displaystyle {\mathsf{\Xi }}_2\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right]\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu }\hfill \\ +\left.{\displaystyle {\int }_1^e\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right]\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right\}\hfill \\ {\displaystyle ={\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\forall \mathsf{\xi }\in [1,e],}\hfill \end{array}$$

where the functions $g_1$, $g_{1i},i=1,\dots ,a,$ $g_{2j},j=1,\dots ,c$, $g_{3k},k=1,\dots ,b$, $g_{4\iota },\iota =1,\dots ,d$ are given by (21) and the functions ${\mathcal{G}}_1,{\mathcal{G}}_2$ are given by (20).

Without presenting the computations for the function v, we obtain for it the relation

$$\begin{array}{c}{\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\forall \mathsf{\xi }\in [1,e],}\hfill \end{array}$$

where the functions ${\mathcal{G}}_3,{\mathcal{G}}_4$ are given by (20). Therefore, we deduce the relations (19). □

Lemma 3. 

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

(a)

$g_1(\mathsf{\xi },\nu )\le g_1(e,\nu )$, $g_2(\mathsf{\xi },\nu )\le g_2(e,\nu )$;

(b)

$g_1(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{g}_1(e,\nu )$, $g_2(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}{g}_2(e,\nu )$;

(c)

$g_1(\mathsf{\xi },\nu )\le {\displaystyle \frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}$, $g_2(\mathsf{\xi },\nu )\le {\displaystyle \frac{{\left(\ln \mathsf{\xi }\right)}^{\beta -1}}{\mathsf{\Gamma }\left(\beta \right)}}$;

(d)

$g_{1i}(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{g}_{1i}(e,\nu )$, $i=1,\dots ,a$;

(e)

$g_{1i}(\mathsf{\xi },\nu )\le {\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}$, $i=1,\dots ,a$;

(f)

$g_{2j}(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -{\eta }_j-1}{g}_{2j}(e,\nu )$, $j=1,\dots ,c$;

(g)

$g_{2j}(\mathsf{\xi },\nu )\le {\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha -{\eta }_j)}}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\eta }_j-1},$$j=1,\dots ,c$;

(h)

$g_{3k}(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -{\sigma }_k-1}{g}_{3k}(e,\nu )$, $k=1,\dots ,b$;

(i)

$g_{3k}(\mathsf{\xi },\nu )\le {\displaystyle \frac{1}{\mathsf{\Gamma }(\beta -{\sigma }_k)}}{\left(\ln \mathsf{\xi }\right)}^{\beta -{\sigma }_k-1}$, $k=1,\dots ,b$;

(j)

$g_{4\iota }(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -{\theta }_{\iota }-1}{g}_{4\iota }(e,\nu )$, $\iota =1,\dots ,d$;

(k)

$g_{4\iota }(\mathsf{\xi },\nu )\le {\displaystyle \frac{1}{\mathsf{\Gamma }(\beta -{\theta }_{\iota })}}{\left(\ln \mathsf{\xi }\right)}^{\beta -{\theta }_{\iota }-1}$, $\iota =1,\dots ,d$;

for all $(\mathsf{\xi },\nu )\in [1,e]\times [1,e]$;

(l)

They are continuous and nonnegative on $[1,e]\times [1,e]$ and positive on $(1,e)\times (1,e)$.

Proof. 

We show the inequalities $\left(a\right)$–$\left(c\right)$ for the function $g_1$. For $g_2$, we can use similar arguments.

(a)

First, we prove that $g_1$ is a nondecreasing function in the first variable. Indeed, for $1\le \nu \le \mathsf{\xi }\le e$, we find

$$\begin{array}{c}{\displaystyle \frac{\partial g_1}{\partial \mathsf{\xi }}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\left[{\displaystyle (\alpha -1){\left(\ln \mathsf{\xi }\right)}^{\alpha -2}\frac{1}{\mathsf{\xi }}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-(\alpha -1){\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -2}\frac{1}{\mathsf{\xi }}}\right]}\hfill \\ {\displaystyle =\frac{(\alpha -1)}{\mathsf{\xi }\mathsf{\Gamma }\left(\alpha \right)}\left[{\left(\ln \mathsf{\xi }\right)}^{\alpha -2}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -2}\right]}\hfill \\ {\displaystyle \ge \frac{1}{\mathsf{\xi }\mathsf{\Gamma }(\alpha -1)}\left[{\left(\ln \mathsf{\xi }\right)}^{\alpha -2}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -2}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\alpha -2}\right]}\hfill \\ {\displaystyle =\frac{1}{\mathsf{\xi }\mathsf{\Gamma }(\alpha -1)}\left[{(\ln \mathsf{\xi }-\ln \mathsf{\xi }\ln \nu )}^{\alpha -2}-{(\ln \mathsf{\xi }-\ln \nu )}^{\alpha -2}\right]\ge 0.}\hfill \end{array}$$

Then,

$$g_1(\mathsf{\xi },\nu )\le g_1(e,\nu )={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\left[{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -1}\right],\forall \mathsf{\xi },\nu \in [1,e].$$

For $1\le \mathsf{\xi }\le \nu \le e$, we obtain

$${\displaystyle \frac{\partial g_1}{\partial \mathsf{\xi }}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\alpha -1)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -2}\frac{1}{\mathsf{\xi }}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\ge 0.}$$

So,

$${\displaystyle g_1(\mathsf{\xi },\nu )\le g_1(\nu ,\nu )=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \nu \right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1},\forall \mathsf{\xi },\nu \in [1,e].}$$

Hence, we deduce that $g_1(\mathsf{\xi },\nu )\le g_1(e,\nu )$ for all $\mathsf{\xi },\nu \in [1,e]$.

(b)

For $1\le \nu \le \mathsf{\xi }\le e$ and $\mathsf{\xi }\ne 1$, we have

$$\begin{array}{c}{\displaystyle g_1(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\left[{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left({\displaystyle \frac{\ln \mathsf{\xi }-\ln \nu }{\ln \mathsf{\xi }}}\right)}^{\alpha -1}\right]}\hfill \\ {\displaystyle =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\left[{(1-\ln \nu )}^{\alpha -\varsigma -1}-{\left({\displaystyle 1-\frac{\ln \nu }{\ln \mathsf{\xi }}}\right)}^{\alpha -1}\right]}\hfill \\ {\displaystyle \ge \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\left[{(1-\ln \nu )}^{\alpha -\varsigma -1}-{(1-\ln \nu )}^{\alpha -1}\right]}\hfill \\ {\displaystyle =\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\left[1-{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\varsigma }\right]}\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{g}_1(e,\nu ).\hfill \end{array}$$

For $\mathsf{\xi }=\nu =1$, we obtain the inequality $g_1(1,1)=0\ge 0={\left(\ln 1\right)}^{\alpha -1}{g}_1(e,1)$.

For $1\le \mathsf{\xi }\le \nu \le e$, we find

$${\displaystyle g_1(\mathsf{\xi },\nu )\ge \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\varsigma }\right)={\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{g}_1(e,s).}$$

So, we deduce $g_1(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{g}_1(e,\nu )$ for all $\mathsf{\xi },\nu \in [1,e]$.

(c)

For all $\mathsf{\xi },\nu \in [1,e]$ we have

$${\displaystyle g_1(\mathsf{\xi },\nu )\le \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\le \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}.}$$

Next, we prove the properties (d)–(e). The properties (f)–(k) follow similar arguments.

(d)

For $1\le \nu \le \mathsf{\xi }\le e$ and $\mathsf{\xi }\ne 1$, we obtain

$$\begin{array}{c}{\displaystyle g_{1i}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}\left[{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}-{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \frac{\ln \mathsf{\xi }-\ln \nu }{\ln \mathsf{\xi }}}\right)}^{\alpha -{\varrho }_i-1}\right]}\hfill \\ {\displaystyle \ge \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}\left[{(1-\ln \nu )}^{\alpha -\varsigma -1}-{(1-\ln \nu )}^{\alpha -{\varrho }_i-1}\right]}\hfill \\ {\displaystyle =\frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\varsigma -{\varrho }_i}\right)}\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{g}_{1i}(e,\nu )\ge 0.\hfill \end{array}$$

For $\mathsf{\xi }=\nu =1$, we have the inequality $g_{1i}(1,1)\ge {\left(\ln 1\right)}^{\alpha -{\varrho }_i-1}{g}_{1i}(e,1)$.

If $1\le \mathsf{\xi }\le \nu \le e$, we find

$$\begin{array}{c}{\displaystyle g_{1i}(\mathsf{\xi },\nu )\ge \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\left(1-{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\varsigma -{\varrho }_i}\right)}\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{g}_{1i}(e,\nu ).\hfill \end{array}$$

Hence, we deduce that $g_{1i}(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{g}_{1i}(e,\nu )$ for all $\mathsf{\xi },\nu \in [1,e]$.

(e)

We have

$${\displaystyle g_{1i}(\mathsf{\xi },\nu )\le \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{\alpha -\varsigma -1}\le \frac{1}{\mathsf{\Gamma }(\alpha -{\varrho }_i)}{\left(\ln \mathsf{\xi }\right)}^{\alpha -{\varrho }_i-1},\forall \mathsf{\xi },\nu \in [1,e].}$$

The last property (l) follows from the definitions of these functions and the properties (b), (d), (f), (h), and (j). □

Lemma 4. 

If ${\mathsf{\Xi }}_1\ge 0$, ${\mathsf{\Xi }}_4\ge 0$, $\Delta >0$ and the functions $H_i,i=1,\dots ,a$, $K_j,j=1,\dots ,b$, $P_k,k=1,\dots ,c$, and $Q_{\iota },\iota =1,\dots ,d$ are nondecreasing, then the Green functions ${\mathcal{G}}_i,i=1,\dots ,4$ satisfy the following inequalities:

(a)

${\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu )\le {\mathcal{G}}_1(\mathsf{\xi },\nu )\le {\mathcal{G}}_1(e,\nu )$,

(b)

${\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu )={\mathcal{G}}_2(\mathsf{\xi },\nu )\le {\mathcal{G}}_2(e,\nu )$,

(c)

${\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\mathcal{G}}_3(e,\nu )={\mathcal{G}}_3(\mathsf{\xi },\nu )\le {\mathcal{G}}_3(e,\nu )$,

(d)

${\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\mathcal{G}}_4(e,\nu )\le {\mathcal{G}}_4(\mathsf{\xi },\nu )\le {\mathcal{G}}_4(e,\nu )$,

for all $\mathsf{\xi },\nu \in [1,e]$.

Proof. 

By using the assumptions of this lemma, we see that ${\mathsf{\Xi }}_1>0$, ${\mathsf{\Xi }}_4>0$, ${\mathsf{\Xi }}_2\ge 0$, and ${\mathsf{\Xi }}_3\ge 0$.

(a)–(d) Based on Lemma 3, for all $\mathsf{\xi },\nu \in [1,e]$, we obtain

$$\begin{array}{c}{\displaystyle {\mathcal{G}}_1(\mathsf{\xi },\nu )\le g_1(e,\nu )+\frac{1}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right]}\hfill \\ ={\mathcal{G}}_1(e,\nu ),\hfill \\ {\displaystyle {\mathcal{G}}_1(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{g}_1(e,\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right]}\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu ),\hfill \\ {\displaystyle {\mathcal{G}}_2(\mathsf{\xi },\nu )\le \frac{1}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_4\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_2\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right]}\hfill \\ ={\mathcal{G}}_2(e,\nu ),\hfill \\ {\mathcal{G}}_2(\mathsf{\xi },\nu )={\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu ),\hfill \\ {\displaystyle {\mathcal{G}}_3(\mathsf{\xi },\nu )\le \frac{1}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_3\sum _{i=1}^a{\int }_1^e{g}_{1i}(\rho ,\nu )d{H}_i\left(\rho \right)+{\mathsf{\Xi }}_1\sum _{i=1}^c{\int }_1^e{g}_{2i}(\rho ,\nu )d{P}_i\left(\rho \right)}\right]}\hfill \\ ={\mathcal{G}}_3(e,\nu ),\hfill \\ {\mathcal{G}}_3(\mathsf{\xi },\nu )={\left(\ln t\right)}^{\beta -1}{\mathcal{G}}_3(e,\nu ),\hfill \\ {\displaystyle {\mathcal{G}}_4(\mathsf{\xi },\nu )\le g_2(e,\nu )+\frac{1}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_3\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_1\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right]}\hfill \\ ={\mathcal{G}}_4(e,\nu ),\hfill \\ {\displaystyle {\mathcal{G}}_4(\mathsf{\xi },\nu )\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}{g}_2(e,\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{\beta -1}}{\Delta }\left[{\displaystyle {\mathsf{\Xi }}_3\sum _{i=1}^b{\int }_1^e{g}_{3i}(\rho ,\nu )d{K}_i\left(\rho \right)+{\mathsf{\Xi }}_1\sum _{i=1}^d{\int }_1^e{g}_{4i}(\rho ,\nu )d{Q}_i\left(\rho \right)}\right]}\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\mathcal{G}}_4(e,\nu ).\hfill \end{array}$$

□

Lemma 5. 

If ${\mathsf{\Xi }}_1\ge 0$, ${\mathsf{\Xi }}_4\ge 0$, $\Delta >0$, the functions $H_i,i=1,\dots ,a$, $K_j,j=1,\dots ,b$, $P_k,k=1,\dots ,c$, and $Q_{\iota },\iota =1,\dots ,d$ are nondecreasing and $\mathfrak{h}\left(\mathsf{\xi }\right)\ge 0$, $\mathfrak{k}\left(\mathsf{\xi }\right)\ge 0$ for all $\mathsf{\xi }\in [1,e]$, then the solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right),\mathsf{\xi }\in [1,e]$, of Problem (8),(2) satisfies the inequalities $\mathfrak{u}\left(\mathsf{\xi }\right)\ge 0$, $\mathfrak{v}\left(\mathsf{\xi }\right)\ge 0$ for all $\mathsf{\xi }\in [1,e]$ and $\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\mathfrak{u}\left(\zeta \right)$, $\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\mathfrak{v}\left(\zeta \right)$ for all $\mathsf{\xi },\zeta \in [1,e]$.

Proof. 

It is easy to verify that $\mathfrak{u}\left(\mathsf{\xi }\right)\ge 0$ and $\mathfrak{v}\left(\mathsf{\xi }\right)\ge 0$ for all $\mathsf{\xi }\in [1,e]$. In addition, by using Lemma 4, we find

$$\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\forall \mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)\ge {\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\left({\displaystyle {\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ \ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\mathfrak{u}\left(\zeta \right),\forall \mathsf{\xi },\zeta \in [1,e],\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu ,\forall \mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)\ge {\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\mathcal{G}}_3(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\left(\ln \mathsf{\xi }\right)}^{\beta -1}{\mathcal{G}}_4(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\hfill \\ ={\left(\ln \mathsf{\xi }\right)}^{\beta -1}\left({\displaystyle {\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{\mathfrak{h}\left(\nu \right)}{\nu }d\nu +{\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{\mathfrak{k}\left(\nu \right)}{\nu }d\nu }\right)\hfill \\ \ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\mathfrak{v}\left(\zeta \right),\forall \mathsf{\xi },\zeta \in [1,e].\hfill \end{array}$$

□

Remark 1. 

Under assumptions of Lemma 5, we obtain $\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\parallel \mathfrak{u}\parallel$ and $\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\parallel \mathfrak{v}\parallel$ for all $\mathsf{\xi }\in [1,e]$, where $\parallel ·\parallel$ is the supremum norm in the space $C\left(\right[1,e\left]\right)$. In addition, by Lemma 4, for $\omega \in (1,e)$, we find

$$\begin{array}{c}{\displaystyle \underset{\mathsf{\xi }\in [\omega ,e]}{min}{\mathcal{G}}_1(\mathsf{\xi },\nu )\ge {\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu ),\forall \nu \in [1,e],}\hfill \\ {\displaystyle \underset{\mathsf{\xi }\in [\omega ,e]}{min}{\mathcal{G}}_2(\mathsf{\xi },\nu )={\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu ),\forall \nu \in [1,e],}\hfill \\ {\displaystyle \underset{\mathsf{\xi }\in [\omega ,e]}{min}{\mathcal{G}}_3(\mathsf{\xi },\nu )={\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_3(e,\nu ),\forall \nu \in [1,e],}\hfill \\ {\displaystyle \underset{\mathsf{\xi }\in [\omega ,e]}{min}{\mathcal{G}}_4(\mathsf{\xi },\nu )\ge {\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_4(e,\nu ),\forall \nu \in [1,e],}\hfill \end{array}$$

and so, by Lemma 5, we deduce

$${\displaystyle \underset{\mathsf{\xi }\in [\omega ,e]}{min}\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \omega \right)}^{\alpha -1}\parallel \mathfrak{u}\parallel ,\underset{\mathsf{\xi }\in [\omega ,e]}{min}\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \omega \right)}^{\beta -1}\parallel \mathfrak{v}\parallel .}$$

Lemma 6. 

If $z\in C\left(\right[1,e\left]\right)$, then for $\kappa >0$, we have

$${\displaystyle |{}^{H}I_{1+}^{\kappa }z\left(\mathsf{\xi }\right)|\le \frac{\parallel z\parallel }{\mathsf{\Gamma }(\kappa +1)},\forall \mathsf{\xi }\in [1,e].}$$

Proof. 

We obtain

$$\begin{array}{c}{\displaystyle |{(}^H{I}_{1+}^{\kappa }z)\left(\mathsf{\xi }\right)| \le \frac{\parallel z\parallel }{\mathsf{\Gamma }\left(\kappa \right)}{\int }_1^{\mathsf{\xi }}{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{\kappa -1}\frac{1}{\nu }d\nu =\frac{{\parallel z\parallel \left(\ln \mathsf{\xi }\right)}^{\kappa }}{\mathsf{\Gamma }(\kappa +1)}\le \frac{\parallel z\parallel }{\mathsf{\Gamma }(\kappa +1)},\forall \mathsf{\xi }\in [1,e].}\hfill \end{array}$$

□

3. Existence of Positive Solutions

We present in this section the theorems devoted to the existence of at least one positive solution for our Problem (1),(2).

We give firstly the main assumptions:

(I1)

$\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 3$; $\beta \in (m-1,m]$, $m\in \mathbb{N}$, $m\ge 3$; $a,b,c,d\in \mathbb{N}$; $0\le {\varrho }_i\le \varsigma <\alpha -1$, $i=1,\dots ,a$; $0\le {\eta }_k\le \varsigma$, $k=1,\dots ,c$; $\varsigma \ge 1$; $0\le {\sigma }_j\le \vartheta <\beta -1$, $j=1,\dots ,b$; $0\le {\theta }_{\iota }\le \vartheta$, $\iota =1,\dots ,d$, $\vartheta \ge 1$; ${\gamma }_i,{\delta }_i>0,i=1,2$; $H_i,i=1,\dots ,a$, $K_j,j=1,\dots ,b$, $P_k,k=1,\dots ,c$, $Q_{\iota },\iota =1,\dots ,d$, are nondecreasing functions; ${\mathsf{\Xi }}_1\ge 0$, ${\mathsf{\Xi }}_4\ge 0$, $\Delta >0$, (given by (9)).

(I2)

The functions $\mathfrak{f},\mathfrak{g}:[1,e]\times {\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ are continuous.

By Lemma 2, Problem (1),(2) can be expressed equivalently as the system of integral equations given by

$$\left\{\begin{array}{c}{\displaystyle \mathfrak{u}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )\mathfrak{f}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )\mathfrak{g}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu },\mathsf{\xi }\in [1,e],}\hfill \\ {\displaystyle \mathfrak{v}\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu )\mathfrak{f}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu )\mathfrak{g}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu },\mathsf{\xi }\in [1,e].}\hfill \end{array}\right.$$

(23)

We consider the Banach space $\mathcal{X}=C\left(\right[1,e\left]\right)$ with the supremum norm $\parallel \mathfrak{u}\parallel ={sup}_{\rho \in [1,e]}\left|\mathfrak{u}\left(\rho \right)\right|$ and the Banach space $\mathcal{Z}=\mathcal{X}\times \mathcal{X}$ with the norm ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}=\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel$. We also consider the cone $\mathcal{S}\subset \mathcal{Z}$ given by

$$\mathcal{S}=\{(\mathfrak{u},\mathfrak{v})\in \mathcal{Z},\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\parallel \mathfrak{u}\parallel ,\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\parallel \mathfrak{v}\parallel ,\forall \mathsf{\xi }\in [1,e]\}.$$

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

$$\begin{array}{c}{\displaystyle {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )\mathfrak{f}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )\mathfrak{g}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu },}\hfill \\ {\displaystyle {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)={\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu )\mathfrak{f}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu )\mathfrak{g}\left(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)\right)\frac{d\nu }{\nu },}\hfill \end{array}$$

for all $\mathsf{\xi }\in [1,e]$ and $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$.

We see that $(\mathfrak{u},\mathfrak{v})$ is a positive solution of system (23) (or, equivalently, to Problem (1),(2)) if and only if it serves as a fixed point of operator $\mathfrak{L}$. Thus, in what follows, we analyze the existence of fixed points of $\mathfrak{L}$.

Under assumptions $\left(I1\right)$, $\left(I2\right)$, by using standard methods, we deduce that operator $\mathfrak{L}$ is completely continuous. In addition, by Lemma 5 and Remark 1, we find

$${\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel ,{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel ,$$

for all $\mathsf{\xi }\in [1,e]$ and $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$, that is, $\mathfrak{L}\left(\mathcal{S}\right)\subset \mathcal{S}$.

For $\omega \in (1,e)$, we introduce the constants

$$\begin{array}{c}{\displaystyle A_1={\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu },A_2={\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu },A_3={\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu },A_4={\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu },}\hfill \\ C_1=A_1+A_3,C_2=A_2+A_4,\hfill \\ {\displaystyle {\tilde{A}}_1={\int }_{\omega }^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu },{\tilde{A}}_2={\int }_{\omega }^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu },{\tilde{A}}_3={\int }_{\omega }^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu },{\tilde{A}}_4={\int }_{\omega }^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu },}\hfill \\ {\tilde{C}}_1={\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_1+{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_3,{\tilde{C}}_2={\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_2+{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_4.\hfill \end{array}$$

We remark that $A_i\ge 0$, ${\tilde{A}}_i\ge 0$ for i = 2,3 and $A_j>0$, ${\tilde{A}}_j>0$ for $j=1,4$, $C_k>0$, ${\tilde{C}}_k>0$ for $k=1,2$. We also denote by ${\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)=\mathfrak{f}(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right))$ and ${\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)=\mathfrak{g}(\nu ,\mathfrak{u}\left(\nu \right),\mathfrak{v}\left(\nu \right){,}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right){,}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right))$ for $\nu \in [1,e]$.

Our first theorem related to the existence and uniqueness of positive solutions of Problem (1),(2) is based on the Banach contraction mapping principle.

Theorem 1. 

We assume that $\left(I1\right)$ and $\left(I2\right)$ hold. In addition, we suppose that there exist continuous functions $X_i,Y_i:[1,e]\to {\mathbb{R}}_{+}$, $i=1.\dots ,4$, such that

$$\begin{array}{c}{\displaystyle |\mathfrak{f}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)-\mathfrak{f}(\mathsf{\xi },{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3,{\mathrm{y}}_4)|\le \sum _{i=1}^4{X}_i\left(\mathsf{\xi }\right)|{\mathrm{x}}_i-{\mathrm{y}}_i|,}\hfill \\ {\displaystyle |\mathfrak{g}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)-\mathfrak{g}(\mathsf{\xi },{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3,{\mathrm{y}}_4)|\le \sum _{i=1}^4{Y}_i\left(\mathsf{\xi }\right)|{\mathrm{x}}_i-{\mathrm{y}}_i|,}\hfill \end{array}$$

(24)

for all $\mathsf{\xi }\in [1,e]$ and ${\mathrm{x}}_i,{\mathrm{y}}_i\in {\mathbb{R}}_{+},i=1,\dots ,4$. If

$${\mathsf{{\rm Y}}}_0=max\{{\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2\}<1,$$

(25)

where

$$\begin{array}{c}{\mathsf{{\rm Y}}}_1=(A_1+A_3)\left({\displaystyle x_1^{*}+\frac{x_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle y_1^{*}+\frac{y_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right),\hfill \\ {\mathsf{{\rm Y}}}_2=(A_1+A_3)\left({\displaystyle x_2^{*}+\frac{x_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle y_2^{*}+\frac{y_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right),\hfill \\ {\displaystyle x_i^{*}=\underset{\mathsf{\xi }\in [1,e]}{sup}{X}_i\left(\mathsf{\xi }\right),y_i^{*}=\underset{\mathsf{\xi }\in [1,e]}{sup}{Y}_i\left(\mathsf{\xi }\right),i=1,\dots ,4,}\hfill \end{array}$$

then the boundary value problem (1),(2) has a unique positive solution $({\mathfrak{u}}^{*}\left(\mathsf{\xi }\right),{\mathfrak{v}}^{*}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [1,e]$. In addition, for any initial point $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \mathcal{S}$, the sequence ${\left(({\mathfrak{u}}_n,{\mathfrak{v}}_n)\right)}_{n\ge 0}$, defined by $({\mathfrak{u}}_n,{\mathfrak{v}}_n)=\mathfrak{L}\left(({\mathfrak{u}}_{n-1},{\mathfrak{v}}_{n-1})\right)$ for $n\ge 1$, converges to $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})$ as $n\to \infty$. Moreover, the error estimate is given by the inequality

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

(26)

Proof. 

By using the inequalities (24), Lemma 4, and Lemma 6, for any $({\mathfrak{u}}_1,{\mathfrak{v}}_1),({\mathfrak{u}}_2,{\mathfrak{v}}_2)\in \mathcal{S}$, we obtain

$$\begin{array}{c}|{\mathfrak{L}}_1({\mathfrak{u}}_1,{\mathfrak{v}}_1)\left(\mathsf{\xi }\right)-{\mathfrak{L}}_1({\mathfrak{u}}_2,{\mathfrak{v}}_2)\left(\mathsf{\xi }\right)|\hfill \\ =\left|{\displaystyle {\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{{\mathfrak{u}}_1{\mathfrak{v}}_1}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{{\mathfrak{u}}_1{\mathfrak{v}}_1}\left(\nu \right)\frac{d\nu }{\nu }}\right.\hfill \\ -\left.{\displaystyle {\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{{\mathfrak{u}}_2{\mathfrak{v}}_2}\left(\nu \right)\frac{d\nu }{\nu }-{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{{\mathfrak{u}}_2{\mathfrak{v}}_2}\left(\nu \right)\frac{d\nu }{\nu }}\right|\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu )|{\mathfrak{F}}_{{\mathfrak{u}}_1{\mathfrak{v}}_1}\left(\nu \right)-{\mathfrak{F}}_{{\mathfrak{u}}_2{\mathfrak{v}}_2}\left(\nu \right)|\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu )|{\mathfrak{G}}_{{\mathfrak{u}}_1{\mathfrak{v}}_1}\left(\nu \right)-{\mathfrak{G}}_{{\mathfrak{u}}_2{\mathfrak{v}}_2}\left(\nu \right)|\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(e,\nu )\left[X_1\left(\nu \right)|{\mathfrak{u}}_1\left(\nu \right)-{\mathfrak{u}}_2\left(\nu \right)|+X_2\left(\nu \right)|{\mathfrak{v}}_1\left(\nu \right)-{\mathfrak{v}}_2\left(\nu \right)|\right.}\hfill \\ {\displaystyle +\left.X_3{\left(\nu \right)|}^H{I}_{1+}^{{\gamma }_1}{\mathfrak{u}}_1\left(\nu \right){-}^H{I}_{1+}^{{\gamma }_1}{\mathfrak{u}}_2\left(\nu \right)|+X_4\left(\nu \right){|}^H{I}_{1+}^{{\delta }_1}{\mathfrak{v}}_1\left(\nu \right){-}^H{I}_{1+}^{{\delta }_1}{\mathfrak{v}}_2\left(\nu \right)|\right]\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_2(e,\nu )\left[Y_1\left(\nu \right)|{\mathfrak{u}}_1\left(\nu \right)-{\mathfrak{u}}_2\left(\nu \right)|+Y_2\left(\nu \right)|{\mathfrak{v}}_1\left(\nu \right)-{\mathfrak{v}}_2\left(\nu \right)|\right.}\hfill \\ {\displaystyle +\left.Y_3{\left(\nu \right)|}^H{I}_{1+}^{{\gamma }_2}{\mathfrak{u}}_1\left(\nu \right){-}^H{I}_{1+}^{{\gamma }_2}{\mathfrak{u}}_2\left(\nu \right)|+Y_4\left(\nu \right){|}^H{I}_{1+}^{{\delta }_2}{\mathfrak{v}}_1\left(\nu \right){-}^H{I}_{1+}^{{\delta }_2}{\mathfrak{v}}_2\left(\nu \right)|\right]\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(e,\nu )\left[X_1\left(\nu \right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +X_2\left(\nu \right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \right.}\hfill \\ {\displaystyle +\left.{\displaystyle X_3\left(\nu \right)\frac{1}{\mathsf{\Gamma }({\gamma }_1+1)}\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +X_4\left(\nu \right)\frac{1}{\mathsf{\Gamma }({\delta }_1+1)}\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel }\right]\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_2(e,\nu )\left[Y_1\left(\nu \right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +Y_2\left(\nu \right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \right.}\hfill \\ {\displaystyle \left.{\displaystyle +Y_3\left(\nu \right)\frac{1}{\mathsf{\Gamma }({\gamma }_2+1)}\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel +Y_4\left(\nu \right)\frac{1}{\mathsf{\Gamma }({\delta }_2+1)}\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel }\right]\frac{d\nu }{\nu },\forall \mathsf{\xi }\in [1,e].}\hfill \end{array}$$

So, we deduce

$$\begin{array}{c}|{\mathfrak{L}}_1({\mathfrak{u}}_1,{\mathfrak{v}}_1)\left(\mathsf{\xi }\right)-{\mathfrak{L}}_1({\mathfrak{u}}_2,{\mathfrak{v}}_2)\left(\mathsf{\xi }\right)|\hfill \\ {\displaystyle \le \left({\displaystyle x_1^{*}+\frac{x_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel {\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\left({\displaystyle x_2^{*}+\frac{x_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel {\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\left({\displaystyle y_1^{*}+\frac{y_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel {\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\left({\displaystyle y_2^{*}+\frac{y_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel {\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ =\parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle x_1^{*}+\frac{x_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)A_1+\left({\displaystyle y_1^{*}+\frac{y_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)A_2\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle x_2^{*}+\frac{x_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)A_1+\left({\displaystyle y_2^{*}+\frac{y_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)A_2\right],\forall \mathsf{\xi }\in [1,e].\hfill \end{array}$$

(27)

In a similar manner, we find

$$\begin{array}{c}|{\mathfrak{L}}_2({\mathfrak{u}}_1,{\mathfrak{v}}_1)\left(\mathsf{\xi }\right)-{\mathfrak{L}}_2({\mathfrak{u}}_2,{\mathfrak{v}}_2)\left(\mathsf{\xi }\right)|\hfill \\ \le \parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle x_1^{*}+\frac{x_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)A_3+\left({\displaystyle y_1^{*}+\frac{y_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)A_4\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle x_2^{*}+\frac{x_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)A_3+\left({\displaystyle y_2^{*}+\frac{y_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)A_4\right],\forall \mathsf{\xi }\in [1,e].\hfill \end{array}$$

(28)

Hence, by (27) and (28), we deduce

$$\begin{array}{c}\parallel \mathfrak{L}({\mathfrak{u}}_1,{\mathfrak{v}}_1)-\mathfrak{L}({\mathfrak{u}}_2,{\mathfrak{v}}_2){\parallel }_{\mathcal{Z}}\hfill \\ \le \parallel {\mathfrak{u}}_1-{\mathfrak{u}}_2\parallel \left[\left({\displaystyle x_1^{*}+\frac{x_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)(A_1+A_3)+\left({\displaystyle y_1^{*}+\frac{y_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)(A_2+A_4)\right]\hfill \\ +\parallel {\mathfrak{v}}_1-{\mathfrak{v}}_2\parallel \left[\left({\displaystyle x_2^{*}+\frac{x_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)(A_1+A_3)+\left({\displaystyle y_2^{*}+\frac{y_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)(A_2+A_4)\right]\hfill \\ \le max\{{\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2\}\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)-({\mathfrak{u}}_2,{\mathfrak{v}}_2){\parallel }_{\mathcal{Z}}={\mathsf{{\rm Y}}}_0{\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1)-({\mathfrak{u}}_2,{\mathfrak{v}}_2)\parallel }_{\mathcal{Z}}.\hfill \end{array}$$

Based on condition (25), it can be concluded that the operator $\mathfrak{L}$ satisfies the contraction mapping property. Thus, by applying the Banach fixed-point theorem, it can be deduced that $\mathfrak{L}$ possesses a single fixed point $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})\in \mathcal{S}$, which corresponds to the unique positive solution of Problem (1),(2). Moreover, for any $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \mathcal{S}$, the sequence ${\left(({\mathfrak{u}}_n,{\mathfrak{v}}_n)\right)}_{n\ge 0}$, defined as $({\mathfrak{u}}_n,{\mathfrak{v}}_n)=\mathfrak{L}({\mathfrak{u}}_{n-1},{\mathfrak{v}}_{n-1})$ for $n\ge 1$, converges to $({\mathfrak{u}}^{*},{\mathfrak{v}}^{*})$ as $n\to \infty$. Finally, the error estimate (26) can be obtained from the proof of the Banach theorem. □

Next, we examine the existence of positive solutions for Problem (1),(2). In the following result, we employ the Schauder fixed-point theorem (see [23]).

Theorem 2. 

Assume that $\left(I1\right)$ and $\left(I2\right)$ hold. In addition, we suppose that there exist continuous functions $U_i,V_i:[1,e]\to {\mathbb{R}}_{+}$, $i=1,\dots ,5$, such that

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)\le \sum _{i=1}^4{U}_i\left(\mathsf{\xi }\right){\mathrm{x}}_i+U_5\left(\mathsf{\xi }\right),\mathfrak{g}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)\le \sum _{i=1}^4{V}_i\left(\mathsf{\xi }\right){\mathrm{x}}_i+V_5\left(\mathsf{\xi }\right),}\hfill \end{array}$$

(29)

for all $\mathsf{\xi }\in [1,e]$ and ${\mathrm{x}}_i\in {\mathbb{R}}_{+}$, $i=1,\dots ,4$. If

$${\mathsf{\Lambda }}_0=max\{{\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2\}<1,$$

(30)

where

$$\begin{array}{c}{\mathsf{\Lambda }}_1=(A_1+A_3)\left({\displaystyle u_1^{*}+\frac{u_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle v_1^{*}+\frac{v_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right),\hfill \\ {\mathsf{\Lambda }}_2=(A_1+A_3)\left({\displaystyle u_2^{*}+\frac{u_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle v_2^{*}+\frac{v_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right),\hfill \\ {\displaystyle u_i^{*}=\underset{\mathsf{\xi }\in [1,e]}{sup}{U}_i\left(\mathsf{\xi }\right),v_i^{*}=\underset{\mathsf{\xi }\in [1,e]}{sup}{V}_i\left(\mathsf{\xi }\right),i=1,\dots ,4,}\hfill \end{array}$$

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

Proof. 

We consider a positive number $r_0$ that satisfies the condition

$${\displaystyle r_0\ge \frac{(A_1+A_3)u_5^{*}+(A_2+A_4)v_5^{*}}{1-{\mathsf{\Lambda }}_0},}$$

where $u_5^{*}={sup}_{\mathsf{\xi }\in [1,e]}{U}_5\left(\mathsf{\xi }\right)$, $v_5^{*}={sup}_{\mathsf{\xi }\in [1,e]}{V}_5\left(\mathsf{\xi }\right)$. We define the set $\mathsf{\Omega }=\{(\mathfrak{u},\mathfrak{v})\in \mathcal{S},\parallel (\mathfrak{u},\mathfrak{v}){\parallel }_{\mathcal{Z}} \le r_0\}$.

First, we prove $\mathfrak{L}\left(\mathsf{\Omega }\right)\subset \mathsf{\Omega }$. For this, let $(\mathfrak{u},\mathfrak{v})\in \mathsf{\Omega }$, that is ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le r_0$, or $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le r_0$, which gives us $\parallel \mathfrak{u}\parallel \le r_0$ and $\parallel \mathfrak{v}\parallel \le r_0$. Then, by using the inequalities (29), Lemma 4, and Lemma 6, we find

$$\begin{array}{c}{\displaystyle {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\le {\int }_1^e{\mathcal{G}}_1(e,\nu )\left(U_1\left(\nu \right)\mathfrak{u}\left(\nu \right)+U_2\left(\nu \right)\mathfrak{v}\left(\nu \right)+U_3{\left(\nu \right)}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right)+U_4{\left(\nu \right)}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)+U_5\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_2(e,\nu )\left(V_1\left(\nu \right)\mathfrak{u}\left(\nu \right)+V_2\left(\nu \right)\mathfrak{v}\left(\nu \right)+V_3{\left(\nu \right)}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right)+V_4{\left(\nu \right)}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)+V_5\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \left({\displaystyle u_1^{*}\parallel \mathfrak{u}\parallel +u_2^{*}\parallel \mathfrak{v}\parallel +u_3^{*}\frac{\parallel \mathfrak{u}\parallel }{\mathsf{\Gamma }({\gamma }_1+1)}+u_4^{*}\frac{\parallel \mathfrak{v}\parallel }{\mathsf{\Gamma }({\delta }_1+1)}+u_5^{*}}\right){\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\left({\displaystyle v_1^{*}\parallel \mathfrak{u}\parallel +v_2^{*}\parallel \mathfrak{v}\parallel +v_3^{*}\frac{\parallel \mathfrak{u}\parallel }{\mathsf{\Gamma }({\gamma }_2+1)}+v_4^{*}\frac{\parallel \mathfrak{v}\parallel }{\mathsf{\Gamma }({\delta }_2+1)}+v_5^{*}}\right){\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ =A_1\left[\left({\displaystyle u_1^{*}+\frac{u_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle u_2^{*}+\frac{u_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)\parallel \mathfrak{v}\parallel +u_5^{*}\right]\hfill \\ +A_2\left[\left({\displaystyle v_1^{*}+\frac{v_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle v_2^{*}+\frac{v_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)\parallel \mathfrak{v}\parallel +v_5^{*}\right],\forall \mathsf{\xi }\in [1,e],\hfill \\ {\displaystyle {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\le {\int }_1^e{\mathcal{G}}_3(e,\nu )\left(U_1\left(\nu \right)\mathfrak{u}\left(\nu \right)+U_2\left(\nu \right)\mathfrak{v}\left(\nu \right)+U_3{\left(\nu \right)}^H{I}_{1+}^{{\gamma }_1}\mathfrak{u}\left(\nu \right)+U_4{\left(\nu \right)}^H{I}_{1+}^{{\delta }_1}\mathfrak{v}\left(\nu \right)+U_5\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e{\mathcal{G}}_4(e,\nu )\left(V_1\left(\nu \right)\mathfrak{u}\left(\nu \right)+V_2\left(\nu \right)\mathfrak{v}\left(\nu \right)+V_3{\left(\nu \right)}^H{I}_{1+}^{{\gamma }_2}\mathfrak{u}\left(\nu \right)+V_4{\left(\nu \right)}^H{I}_{1+}^{{\delta }_2}\mathfrak{v}\left(\nu \right)+V_5\left(\nu \right)\right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \left({\displaystyle u_1^{*}\parallel \mathfrak{u}\parallel +u_2^{*}\parallel \mathfrak{v}\parallel +u_3^{*}\frac{\parallel \mathfrak{u}\parallel }{\mathsf{\Gamma }({\gamma }_1+1)}+u_4^{*}\frac{\parallel \mathfrak{v}\parallel }{\mathsf{\Gamma }({\delta }_1+1)}+u_5^{*}}\right){\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\left({\displaystyle v_1^{*}\parallel \mathfrak{u}\parallel +v_2^{*}\parallel \mathfrak{v}\parallel +v_3^{*}\frac{\parallel \mathfrak{u}\parallel }{\mathsf{\Gamma }({\gamma }_2+1)}+v_4^{*}\frac{\parallel \mathfrak{v}\parallel }{\mathsf{\Gamma }({\delta }_2+1)}+v_5^{*}}\right){\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu }}\hfill \\ =A_3\left[\left({\displaystyle u_1^{*}+\frac{u_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle u_2^{*}+\frac{u_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)\parallel \mathfrak{v}\parallel +u_5^{*}\right]\hfill \\ +A_4\left[\left({\displaystyle v_1^{*}+\frac{v_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle v_2^{*}+\frac{v_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)\parallel \mathfrak{v}\parallel +v_5^{*}\right],\forall \mathsf{\xi }\in [1,e].\hfill \end{array}$$

(31)

Then, by (31), we deduce

$$\begin{array}{c}{\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le (A_1+A_3)\left[\left({\displaystyle u_1^{*}+\frac{u_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle u_2^{*}+\frac{u_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)\parallel \mathfrak{v}\parallel +u_5^{*}\right]\hfill \\ +(A_2+A_4)\left[\left({\displaystyle v_1^{*}+\frac{v_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)\parallel \mathfrak{u}\parallel +\left({\displaystyle v_2^{*}+\frac{v_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)\parallel \mathfrak{v}\parallel +v_5^{*}\right]\hfill \\ =\left[(A_1+A_3)\left({\displaystyle u_1^{*}+\frac{u_3^{*}}{\mathsf{\Gamma }({\gamma }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle v_1^{*}+\frac{v_3^{*}}{\mathsf{\Gamma }({\gamma }_2+1)}}\right)\right]\parallel \mathfrak{u}\parallel \hfill \\ +\left[(A_1+A_3)\left({\displaystyle u_2^{*}+\frac{u_4^{*}}{\mathsf{\Gamma }({\delta }_1+1)}}\right)+(A_2+A_4)\left({\displaystyle v_2^{*}+\frac{v_4^{*}}{\mathsf{\Gamma }({\delta }_2+1)}}\right)\right]\parallel \mathfrak{v}\parallel \hfill \\ +(A_1+A_3)u_5^{*}+(A_2+A_4)v_5^{*}\hfill \\ ={\mathsf{\Lambda }}_1\parallel \mathfrak{u}\parallel +{\mathsf{\Lambda }}_2\parallel \mathfrak{v}\parallel +(A_1+A_3)u_5^{*}+(A_2+A_4)v_5^{*}\hfill \\ \le {\mathsf{\Lambda }}_0{\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}+(A_1+A_3)u_5^{*}+(A_2+A_4)v_5^{*}\hfill \\ \le {\mathsf{\Lambda }}_0{r}_0+(A_1+A_3)u_5^{*}+(A_2+A_4)v_5^{*}\le r_0.\hfill \end{array}$$

So, $\mathfrak{L}\left(\mathsf{\Omega }\right)\subset \mathsf{\Omega }$. Since the operator $\mathfrak{L}$ is completely continuous, we can apply the Schauder fixed-point theorem to conclude the existence of a fixed point $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$, with ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le r_0$. This fixed point serves as a positive solution to Problem (1),(2). □

The next theorem is based on the Guo–Krasnosel’skii fixed-point theorem (see [24]).

Theorem 3. 

Let $\omega \in (1,e)$. We assume that $\left(I1\right)$ and $\left(I2\right)$ hold. In addition, we suppose that there exist two positive constants $R_2>R_1>0$ and the constants ${\lambda }_1\in (0,C_1^{-1}]$, ${\lambda }_2\in (0,C_2^{-1}]$, ${\lambda }_3\in [{\tilde{C}}_1^{-1},\infty )$, ${\lambda }_4\in [{\tilde{C}}_2^{-1},\infty )$ such that

$$(I3)\hspace{1em}\left\{\begin{array}{c}\mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\ge {\displaystyle \frac{{\lambda }_3{R}_1}{2}},\forall \mathsf{\xi }\in [\omega ,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_1,\mathrm{x}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\gamma }_1+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\delta }_1+1)}}\right];\hfill \\ \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\ge {\displaystyle \frac{{\lambda }_4{R}_1}{2}},\forall \mathsf{\xi }\in [\omega ,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_1,\mathrm{x}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\gamma }_2+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\delta }_2+1)}}\right];\hfill \end{array}\right.$$

$$(I4)\hspace{1em}\left\{\begin{array}{c}\mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le {\displaystyle \frac{{\lambda }_1{R}_2}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_2,\mathrm{x}\in \left[0,{\displaystyle \frac{R_2}{\mathsf{\Gamma }({\gamma }_1+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_2}{\mathsf{\Gamma }({\delta }_1+1)}}\right];\hfill \\ \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le {\displaystyle \frac{{\lambda }_2{R}_2}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_2,\mathrm{x}\in \left[0,{\displaystyle \frac{R_2}{\mathsf{\Gamma }({\gamma }_2+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_2}{\mathsf{\Gamma }({\delta }_2+1)}}\right].\hfill \end{array}\right.$$

Then, Problem (1),(2) has at least one positive solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right),\mathsf{\xi }\in [1,e]$ such that $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$ and $R_1\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le R_2$.

Proof. 

We define the set ${\mathsf{\Omega }}_1={\{(\mathfrak{u},\mathfrak{v})\in \mathcal{Z},\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}<R_1\}$. Then, for $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}\cap \partial {\mathsf{\Omega }}_1$, we have $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel =R_1$, so $\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right)\le R_1$ for all $\mathsf{\xi }\in [1,e]$. Because $\parallel \mathfrak{u}\parallel \le R_1$ and $\parallel \mathfrak{v}\parallel \le R_1$, by Lemma 6, we find $|{}^{H}I_{1+}^{{\gamma }_i}\mathfrak{u}\left(\nu \right)|\le {\displaystyle \frac{R_1}{\mathsf{\Gamma }({\gamma }_i+1)}}$ and $|{}^{H}I_{1+}^{{\delta }_i}\mathfrak{v}\left(\nu \right)|\le {\displaystyle \frac{R_1}{\mathsf{\Gamma }({\delta }_i+1)}}$ for all $\nu \in [1,e]$ and $i=1,2$. Then, by Lemma 4 and assumption $\left(I3\right)$, we deduce

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle =\underset{\mathsf{\xi }\in [1,e]}{sup}\left({\displaystyle {\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\right)}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}\left({\displaystyle {\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\right)}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_{\omega }^e{\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_{\omega }^e{\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge \frac{{\lambda }_3{R}_1}{2}{\left(\ln \omega \right)}^{\alpha -1}{\int }_{\omega }^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_4{R}_1}{2}{\left(\ln \omega \right)}^{\alpha -1}{\int }_{\omega }^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_3{R}_1{\tilde{A}}_1}{2}{\left(\ln \omega \right)}^{\alpha -1}+\frac{{\lambda }_4{R}_1{\tilde{A}}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}}\hfill \\ {\displaystyle =\frac{{\lambda }_3{\tilde{A}}_1+{\lambda }_4{\tilde{A}}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}{R}_1,}\hfill \end{array}$$

(32)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle =\underset{\mathsf{\xi }\in [1,e]}{sup}\left({\displaystyle {\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\right)}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}\left({\displaystyle {\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\right)}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_{\omega }^e{\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_3(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_{\omega }^e{\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_4(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge \frac{{\lambda }_3{R}_1}{2}{\left(\ln \omega \right)}^{\beta -1}{\int }_{\omega }^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_4{R}_1}{2}{\left(\ln \omega \right)}^{\beta -1}{\int }_{\omega }^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_3{R}_1{\tilde{A}}_3}{2}{\left(\ln \omega \right)}^{\beta -1}+\frac{{\lambda }_4{R}_1{\tilde{A}}_4}{2}{\left(\ln \omega \right)}^{\beta -1}}\hfill \\ {\displaystyle =\frac{{\lambda }_3{\tilde{A}}_3+{\lambda }_4{\tilde{A}}_4}{2}{\left(\ln \omega \right)}^{\beta -1}{R}_1.}\hfill \end{array}$$

(33)

Then, by (32) and (33), we conclude

$$\begin{array}{c}{\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}=\parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel +\parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel \hfill \\ \ge \left[{\displaystyle \frac{{\lambda }_3{\tilde{A}}_1+{\lambda }_4{\tilde{A}}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}+\frac{{\lambda }_3{\tilde{A}}_3+{\lambda }_4{\tilde{A}}_4}{2}{\left(\ln \omega \right)}^{\beta -1}}\right]R_1\hfill \\ =\left[{\displaystyle \frac{{\lambda }_3({\tilde{A}}_1{\left(\ln \omega \right)}^{\alpha -1}+{\tilde{A}}_3{\left(\ln \omega \right)}^{\beta -1})}{2}+\frac{{\lambda }_4({\tilde{A}}_2{\left(\ln \omega \right)}^{\alpha -1}+{\tilde{A}}_4{\left(\ln \omega \right)}^{\beta -1})}{2}}\right]R_1\hfill \\ =\left({\displaystyle \frac{{\lambda }_3{\tilde{C}}_1}{2}+\frac{{\lambda }_4{\tilde{C}}_2}{2}}\right)R_1\ge \left({\displaystyle \frac{1}{2}+\frac{1}{2}}\right)R_1=R_1,\hfill \end{array}$$

and therefore,

$${\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\ge {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}},\forall (\mathfrak{u},\mathfrak{v})\in \mathcal{S}\cap \partial {\mathsf{\Omega }}_1.$$

(34)

We introduce now the set ${\mathsf{\Omega }}_2={\{(\mathfrak{u},\mathfrak{v})\in \mathcal{S},\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}<R_2\}$. Then, for $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}\cap \partial {\mathsf{\Omega }}_2$, we find $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel =R_2$, so $\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right)\le R_2$ for all $\mathsf{\xi }\in [1,e]$. Therefore, by Lemma 4 and assumption $\left(I4\right)$, we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle \le \underset{\mathsf{\xi }\in [1,e]}{sup}{\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [1,e]}{sup}{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e\underset{\mathsf{\xi }\in [1,e]}{sup}{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [1,e]}{sup}{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \frac{{\lambda }_1{R}_2}{2}{\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_2{R}_2}{2}{\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_1{R}_2}{2}{A}_1+\frac{{\lambda }_2{R}_2}{2}{A}_2=\left({\displaystyle \frac{{\lambda }_1{A}_1}{2}+\frac{{\lambda }_2{A}_2}{2}}\right)R_2,}\hfill \end{array}$$

(35)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle \le \underset{\mathsf{\xi }\in [1,e]}{sup}{\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [1,e]}{sup}{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e\underset{\mathsf{\xi }\in [1,e]}{sup}{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [1,e]}{sup}{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_3(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_4(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \frac{{\lambda }_1{R}_2}{2}{\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_2{R}_2}{2}{\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_1{R}_2}{2}{A}_3+\frac{{\lambda }_2{R}_2}{2}{A}_4=\left({\displaystyle \frac{{\lambda }_1{A}_3}{2}+\frac{{\lambda }_2{A}_4}{2}}\right)R_2.}\hfill \end{array}$$

(36)

Then, by (35) and (36), we conclude

$$\begin{array}{c}{\displaystyle {\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}=\parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel +\parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel \le \frac{{\lambda }_1{A}_1+{\lambda }_2{A}_2}{2}{R}_2+\frac{{\lambda }_1{A}_3+{\lambda }_2{A}_4}{2}{R}_2}\hfill \\ =\left({\displaystyle \frac{{\lambda }_1(A_1+A_3)}{2}+\frac{{\lambda }_2(A_2+A_4)}{2}}\right)R_2\hfill \\ =\left({\displaystyle \frac{{\lambda }_1{C}_1}{2}+\frac{{\lambda }_2{C}_2}{2}}\right)R_2\le \left({\displaystyle \frac{1}{2}+\frac{1}{2}}\right)R_2=R_2,\hfill \end{array}$$

that is,

$${\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}},\forall (\mathfrak{u},\mathfrak{v})\in \mathcal{S}\cap \partial {\mathsf{\Omega }}_2.$$

(37)

The operator $\mathfrak{L}$ is completely continuous. Therefore, by (34) and (37) and the Guo–Krasnosel’skii fixed-point theorem, we deduce that operator $\mathfrak{L}$ has a fixed point $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}\cap ({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1)$, which is a positive solution of Problem (1),(2). This solution satisfies the inequalities $R_1\le {\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le R_2$, $\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\alpha -1}\parallel \mathfrak{u}\parallel$, $\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{\beta -1}\parallel \mathfrak{v}\parallel$ for all $\mathsf{\xi }\in [1,e]$; because $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \ge R_1$, we find $\parallel \mathfrak{u}\parallel >0$ or $\parallel \mathfrak{v}\parallel >0$, that is, $\mathfrak{u}\left(\mathsf{\xi }\right)>0$ for all $\mathsf{\xi }\in (1,e]$ or $\mathfrak{v}\left(\mathsf{\xi }\right)>0$ for all $\mathsf{\xi }\in (1,e]$. □

In the last result, we employ the Leggett–Williams fixed-point theorem (see Theorem 3.3 in [25]), and we prove that Problem (1),(2) has at least three positive solutions.

Theorem 4. 

Let $\omega \in (1,e)$. We assume that $\left(I1\right)$ and $\left(I2\right)$ hold. In addition, we suppose that there exist positive constants $0<R_1<R_2<R_3$ and the constants ${\lambda }_1\in (0,C_1^{-1}]$, ${\lambda }_2\in (0,C_2^{-1}]$, ${\lambda }_3\in [{\tilde{C}}_1^{-1},\infty )$, ${\lambda }_4\in [{\tilde{C}}_2^{-1},\infty )$, ${\lambda }_5\in (0,C_1^{-1}]$, ${\lambda }_6\in (0,C_2^{-1}]$ such that

$$(I5)\hspace{1em}\left\{\begin{array}{c}\mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})<{\displaystyle \frac{{\lambda }_1{R}_1}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_1,\mathrm{x}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\gamma }_1+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\delta }_1+1)}}\right];\hfill \\ \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})<{\displaystyle \frac{{\lambda }_2{R}_1}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_1,\mathrm{x}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\gamma }_2+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_1}{\mathsf{\Gamma }({\delta }_2+1)}}\right];\hfill \end{array}\right.$$

$$(I6)\hspace{1em}\left\{\begin{array}{c}\mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})>{\displaystyle \frac{{\lambda }_3{R}_2}{2}},\forall \mathsf{\xi }\in [\omega ,e],\mathrm{u},\mathrm{v}\ge 0,R_2\le \mathrm{u}+\mathrm{v}\le R_3,\mathrm{x}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\gamma }_1+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\delta }_1+1)}}\right];\hfill \\ \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})>{\displaystyle \frac{{\lambda }_4{R}_2}{2}},\forall \mathsf{\xi }\in [\omega ,e],\mathrm{u},\mathrm{v}\ge 0,R_2\le \mathrm{u}+\mathrm{v}\le R_3,\mathrm{x}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\gamma }_2+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\delta }_2+1)}}\right];\hfill \end{array}\right.$$

$$(I7)\hspace{1em}\left\{\begin{array}{c}\mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le {\displaystyle \frac{{\lambda }_5{R}_3}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_3,\mathrm{x}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\gamma }_1+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\delta }_1+1)}}\right];\hfill \\ \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le {\displaystyle \frac{{\lambda }_6{R}_3}{2}},\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le R_3,\mathrm{x}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\gamma }_2+1)}}\right],\mathrm{y}\in \left[0,{\displaystyle \frac{R_3}{\mathsf{\Gamma }({\delta }_2+1)}}\right].\hfill \end{array}\right.$$

Then, Problem (1),(2) has at least three positive solutions $({\mathfrak{u}}_i\left(\mathsf{\xi }\right),{\mathfrak{v}}_i\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [1,e]$, $i=1,\dots ,3$, such that $({\mathfrak{u}}_i,{\mathfrak{v}}_i)\in \mathcal{S}$, $i=1,\dots ,3$, $\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1){\parallel }_{\mathcal{Z}} < R_1$, $\parallel ({\mathfrak{u}}_3,{\mathfrak{v}}_3){\parallel }_{\mathcal{Z}} > R_1$, ${inf}_{\mathsf{\xi }\in [\omega ,e]}({\mathfrak{u}}_2\left(\mathsf{\xi }\right)+{\mathfrak{v}}_2\left(\mathsf{\xi }\right))>R_2$, and ${inf}_{\mathsf{\xi }\in [\omega ,e]}({\mathfrak{u}}_3\left(\mathsf{\xi }\right)+{\mathfrak{v}}_3\left(\mathsf{\xi }\right))<R_2$.

Proof. 

We show firstly that $\mathfrak{L}:{\overline{\mathcal{S}}}_{R_3}\to {\overline{\mathcal{S}}}_{R_3}$, where ${\overline{\mathcal{S}}}_{R_3}$ is the closure of the set ${\mathcal{S}}_{R_3}={\{(\mathfrak{u},\mathfrak{v})\in \mathcal{S},\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}<R_3\}$. Indeed, for any $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{S}}}_{R_3}$, we have ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le R_3$, and then, $\parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le R_3$ or $0\le \mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right)\le R_3$ for all $\mathsf{\xi }\in [1,e]$. By using $\left(I7\right)$ and Lemma 4, we deduce

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_1(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_2(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \frac{{\lambda }_5{R}_3}{2}{\int }_1^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_6{R}_3}{2}{\int }_1^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_5{R}_3}{2}{A}_1+\frac{{\lambda }_6{R}_3}{2}{A}_2=\left({\displaystyle \frac{{\lambda }_5{A}_1}{2}+\frac{{\lambda }_6{A}_2}{2}}\right)R_3,}\hfill \end{array}$$

(38)

and

$$\begin{array}{c}{\displaystyle \parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel =\underset{\mathsf{\xi }\in [1,e]}{sup}\left|{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)\right|}\hfill \\ {\displaystyle \le {\int }_1^e{\mathcal{G}}_3(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e{\mathcal{G}}_4(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \le \frac{{\lambda }_5{R}_3}{2}{\int }_1^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_6{R}_3}{2}{\int }_1^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_5{R}_3}{2}{A}_3+\frac{{\lambda }_6{R}_3}{2}{A}_4=\left({\displaystyle \frac{{\lambda }_5{A}_3}{2}+\frac{{\lambda }_6{A}_4}{2}}\right)R_3.}\hfill \end{array}$$

(39)

Then, by (38) and (39), we find

$$\begin{array}{c}{\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le \left({\displaystyle \frac{{\lambda }_5(A_1+A_3)}{2}+\frac{{\lambda }_6(A_2+A_4)}{2}}\right)R_3\hfill \\ \le \left({\displaystyle \frac{A_1+A_3}{2C_1}+\frac{A_2+A_4}{2C_2}}\right)R_3=\left({\displaystyle \frac{1}{2}+\frac{1}{2}}\right)R_3=R_3,\forall (\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{S}}}_{R_3}.\hfill \end{array}$$

Therefore, $\mathfrak{L}\left({\overline{\mathcal{S}}}_{R_3}\right)\subset {\overline{\mathcal{S}}}_{R_3}$.

We consider now ${\tilde{R}}_3\in (R_2,R_3)$, and we define the concave nonnegative continuous functional $\mathsf{\Phi }$ on $\mathcal{S}$ by $\mathsf{\Phi }(\mathfrak{u},\mathfrak{v})={inf}_{\mathsf{\xi }\in [\omega ,e]}(\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right)),(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$. We note that $\mathsf{\Phi }(\mathfrak{u},\mathfrak{v})\le \parallel (\mathfrak{u},\mathfrak{v})\parallel$ for all $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{S}}}_{R_3}$.

Next, we verify the conditions $\left(1\right)$–$\left(3\right)$ of Theorem 3.3 from [25], with $E=\mathcal{Z}$, $K=\mathcal{S}$, $A=\mathfrak{L}$, $d=R_1$, $a=R_2$, $c=R_3$, $b={\tilde{R}}_3$, $\alpha =\mathsf{\Phi }$, $K_c={\overline{\mathcal{S}}}_{R_3}$, and $S(\alpha ,a,b)=\mathcal{S}(\mathsf{\Phi },R_2,{\tilde{R}}_3)$.

Firstly, we verify condition $\left(1\right)$. We define the element $({\mathfrak{u}}_0\left(\mathsf{\xi }\right),{\mathfrak{v}}_0\left(\mathsf{\xi }\right))=({\displaystyle \frac{R_2+{\tilde{R}}_3}{4}},{\displaystyle \frac{R_2+{\tilde{R}}_3}{4}})$, $\mathsf{\xi }\in [1,e]$. Because ${\mathfrak{u}}_0\left(\mathsf{\xi }\right)\ge 0$, ${\mathfrak{v}}_0\left(\mathsf{\xi }\right)\ge 0$ for all $\mathsf{\xi }\in [1,e]$, $\parallel ({\mathfrak{u}}_0,{\mathfrak{v}}_0){\parallel }_{\mathcal{Z}}={\displaystyle \frac{R_2+{\tilde{R}}_3}{2}}<{\tilde{R}}_3$, and $\mathsf{\Phi }({\mathfrak{u}}_0,{\mathfrak{v}}_0)={\displaystyle \frac{R_2+{\tilde{R}}_3}{2}}>R_2$, we conclude that $({\mathfrak{u}}_0,{\mathfrak{v}}_0)\in \{(\mathfrak{u},\mathfrak{v});(\mathfrak{u},\mathfrak{v})\in \mathcal{S}(\mathsf{\Phi },R_2,{\tilde{R}}_3),\mathsf{\Phi }(\mathfrak{u},\mathfrak{v})>R_2\}$. Now, let $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}(\mathsf{\Phi },R_2,{\tilde{R}}_3)$, that is, $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}$, $\mathsf{\Phi }(\mathfrak{u},\mathfrak{v})\ge R_2$ and ${\parallel (\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}\le {\tilde{R}}_3$. Then, $\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right)\le {\tilde{R}}_3$ for all $\mathsf{\xi }\in [1,e]$ and ${inf}_{\mathsf{\xi }\in [\omega ,e]}(\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{v}\left(\mathsf{\xi }\right))\ge R_2$. Therefore, by $\left(I6\right)$, Lemma 4 and Remark 1, we deduce

$$\begin{array}{c}{\displaystyle \mathsf{\Phi }\left(\mathfrak{L}(\mathfrak{u},\mathfrak{v})\right)=\underset{\mathsf{\xi }\in [\omega ,e]}{inf}({\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)+{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right))}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)+\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\left(\mathsf{\xi }\right)}\hfill \\ {\displaystyle \ge \underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\int }_1^e{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_1(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_2(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_3(\mathsf{\xi },\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_1^e\underset{\mathsf{\xi }\in [\omega ,e]}{inf}{\mathcal{G}}_4(\mathsf{\xi },\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle \ge {\int }_{\omega }^e{\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_1(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_{\omega }^e{\left(\ln \omega \right)}^{\alpha -1}{\mathcal{G}}_2(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +{\int }_{\omega }^e{\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_3(e,\nu ){\mathfrak{F}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }+{\int }_{\omega }^e{\left(\ln \omega \right)}^{\beta -1}{\mathcal{G}}_4(e,\nu ){\mathfrak{G}}_{\mathfrak{u}\mathfrak{v}}\left(\nu \right)\frac{d\nu }{\nu }}\hfill \\ {\displaystyle >\frac{{\lambda }_3{R}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}{\int }_{\omega }^e{\mathcal{G}}_1(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_4{R}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}{\int }_{\omega }^e{\mathcal{G}}_2(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle +\frac{{\lambda }_3{R}_2}{2}{\left(\ln \omega \right)}^{\beta -1}{\int }_{\omega }^e{\mathcal{G}}_3(e,\nu )\frac{d\nu }{\nu }+\frac{{\lambda }_4{R}_2}{2}{\left(\ln \omega \right)}^{\beta -1}{\int }_{\omega }^e{\mathcal{G}}_4(e,\nu )\frac{d\nu }{\nu }}\hfill \\ {\displaystyle =\frac{{\lambda }_3{R}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_1+\frac{{\lambda }_4{R}_2}{2}{\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_2}\hfill \\ {\displaystyle +\frac{{\lambda }_3{R}_2}{2}{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_3+\frac{{\lambda }_4{R}_2}{2}{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_4}\hfill \\ {\displaystyle \ge \frac{R_2{\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_1}{2{\tilde{C}}_1}+\frac{R_2{\left(\ln \omega \right)}^{\alpha -1}{\tilde{A}}_2}{2{\tilde{C}}_2}+\frac{R_2{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_3}{2{\tilde{C}}_1}+\frac{R_2{\left(\ln \omega \right)}^{\beta -1}{\tilde{A}}_4}{2{\tilde{C}}_2}}\hfill \\ =\left[{\displaystyle \frac{{\tilde{A}}_1{\left(\ln \omega \right)}^{\alpha -1}+{\tilde{A}}_3{\left(\ln \omega \right)}^{\beta -1}}{2{\tilde{C}}_1}+\frac{{\tilde{A}}_2{\left(\ln \omega \right)}^{\alpha -1}+{\tilde{A}}_4{\left(\ln \omega \right)}^{\beta -1}}{2{\tilde{C}}_2}}\right]R_2\hfill \\ =\left({\displaystyle \frac{{\tilde{C}}_1}{2{\tilde{C}}_1}+\frac{{\tilde{C}}_2}{2{\tilde{C}}_2}}\right)R_2=R_2.\hfill \end{array}$$

Therefore, $\mathsf{\Phi }\left(\mathfrak{L}(\mathfrak{u},\mathfrak{v})\right)>R_2$, and we have condition $\left(1\right)$ of Theorem 3.3 from [25].

We verify now condition $\left(2\right)$. For $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{S}}}_{R_1}$, we prove that ${\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}<R_1$. For this, let $(\mathfrak{u},\mathfrak{v})\in {\overline{\mathcal{S}}}_{R_1}$. Using $\left(I5\right)$ and similar arguments than those used in the proof of Equations (38) and (39), we find

$$\parallel {\mathfrak{L}}_1(\mathfrak{u},\mathfrak{v})\parallel <\left({\displaystyle \frac{{\lambda }_1{A}_1}{2}+\frac{{\lambda }_2{A}_2}{2}}\right)R_1,\parallel {\mathfrak{L}}_2(\mathfrak{u},\mathfrak{v})\parallel <\left({\displaystyle \frac{{\lambda }_1{A}_3}{2}+\frac{{\lambda }_2{A}_4}{2}}\right)R_1,$$

and then,

$${\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}<\left({\displaystyle \frac{A_1+A_3}{2C_1}+\frac{A_2+A_4}{2C_2}}\right)R_1=R_1.$$

So, we have condition $\left(2\right)$ of Theorem 3.3 from [25].

Finally, we verify condition $\left(3\right)$, that is, $\mathsf{\Phi }\left(\mathfrak{L}(\mathfrak{u},\mathfrak{v})\right)>R_2$ for all $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}(\mathsf{\Phi },R_2,R_3)$ and ${\parallel \mathcal{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}>{\tilde{R}}_3$. For this, let $(\mathfrak{u},\mathfrak{v})\in \mathcal{S}(\mathsf{\Phi },R_2,R_3)$ and ${\parallel \mathfrak{L}(\mathfrak{u},\mathfrak{v})\parallel }_{\mathcal{Z}}>{\tilde{R}}_3$. By using similar arguments as those from $\left(1\right)$, we obtain that $\mathsf{\Phi }\left(\mathfrak{L}(\mathfrak{u},\mathfrak{v})\right)>R_2$; that is, condition $\left(3\right)$ of Theorem 3.3 from [25] is also satisfied.

The operator $\mathfrak{L}$ is completely continuous. Then, by using the Leggett–Williams fixed-point theorem, we conclude that Problem (1),(2) has at least three positive solutions $({\mathfrak{u}}_i\left(\mathsf{\xi }\right),{\mathfrak{v}}_i\left(\mathsf{\xi }\right)),$ $\mathsf{\xi }\in [1,e]$, with $({\mathfrak{u}}_i,{\mathfrak{v}}_i)\in \mathcal{S}$, $i=1,\dots ,3$, and $\parallel ({\mathfrak{u}}_1,{\mathfrak{v}}_1){\parallel }_{\mathcal{Z}}<R_1$, $\parallel ({\mathfrak{u}}_3,{\mathfrak{v}}_3){\parallel }_{\mathcal{Z}}>R_1$, $\mathsf{\Phi }({\mathfrak{u}}_2,{\mathfrak{v}}_2)={inf}_{\mathsf{\xi }\in [\omega ,e]}({\mathfrak{u}}_2\left(\mathsf{\xi }\right)+{\mathfrak{v}}_2\left(\mathsf{\xi }\right))>R_2$, and ${\displaystyle \mathsf{\Phi }({\mathfrak{u}}_3,{\mathfrak{v}}_3)=\underset{\mathsf{\xi }\in [\omega ,e]}{inf}({\mathfrak{u}}_3\left(\mathsf{\xi }\right)+{\mathfrak{v}}_3\left(\mathsf{\xi }\right))<R_2}$. □

4. Examples

Let $\alpha ={\displaystyle \frac{9}{4}}$, $n=3$, $\beta ={\displaystyle \frac{11}{3}}$, $m=4$, ${\gamma }_1={\displaystyle \frac{15}{26}}$, ${\delta }_1={\displaystyle \frac{31}{7}}$, ${\gamma }_2={\displaystyle \frac{42}{5}}$, ${\delta }_2={\displaystyle \frac{29}{72}}$, $\varsigma ={\displaystyle \frac{8}{7}}$, $\vartheta ={\displaystyle \frac{12}{5}}$, $a=b=c=d=1$, ${\varrho }_1={\displaystyle \frac{4}{9}}$, ${\sigma }_1=0$, ${\eta }_1={\displaystyle \frac{2}{3}}$, ${\theta }_1={\displaystyle \frac{19}{12}}$.

We also consider the functions

$$\begin{gathered} H_1\left(\nu \right)=\left\{\begin{array}{c}1,\nu \in [1,{\displaystyle \frac{3}{2}}),\hfill \\ {\displaystyle \frac{7}{6}},\nu \in [{\displaystyle \frac{3}{2}},2),\hfill \\ {\displaystyle \frac{25}{49}}{(\nu -1)}^{7/5}+{\displaystyle \frac{193}{294}},\nu \in [2,e],\hfill \end{array}\right.K_1\left(\nu \right)=\left\{\begin{array}{c}2,\nu \in [1,2),\hfill \\ {\displaystyle \frac{7}{3}},\nu \in [2,{\displaystyle \frac{5}{2}}),\hfill \\ {\displaystyle \frac{3}{4}}{(\nu -2)}^{4/3}-{\displaystyle \frac{3}{8\sqrt[3]{2}}}+{\displaystyle \frac{7}{3}},\nu \in [{\displaystyle \frac{5}{2}},e],\hfill \end{array}\right. \\ {P}_1\left(\nu \right)=\left\{\begin{array}{c}{\displaystyle \frac{12}{5}}{(\nu -{\displaystyle \frac{1}{3}})}^{5/4},\nu \in [1,{\displaystyle \frac{6}{5}}),\hfill \\ {\displaystyle \frac{52}{25}}{\left({\displaystyle \frac{13}{15}}\right)}^{1/4},\nu \in [{\displaystyle \frac{6}{5}},{\displaystyle \frac{15}{7}}),\hfill \\ {\displaystyle \frac{52}{25}}{\left({\displaystyle \frac{13}{15}}\right)}^{1/4}+{\displaystyle \frac{1}{2}},\nu \in [{\displaystyle \frac{15}{7}},e],\hfill \end{array}\right.Q_1\left(\nu \right)=\left\{\begin{array}{c}{\displaystyle \frac{8}{9}}{(\nu -{\displaystyle \frac{1}{4}})}^{9/11},\nu \in [1,{\displaystyle \frac{4}{3}}),\hfill \\ {\displaystyle \frac{8}{9}}{\left({\displaystyle \frac{13}{12}}\right)}^{9/11},\nu \in [{\displaystyle \frac{4}{3}},e].\hfill \end{array}\right. \end{gathered}$$

We consider the system of Hadamard fractional differential equations

$$\left\{\begin{array}{c}{}^{H}D_{1+}^{9/4}\mathfrak{u}\left(\mathsf{\xi }\right)+\mathfrak{f}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{15/26}\mathfrak{u}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{31/7}\mathfrak{v}\left(\mathsf{\xi }\right))=0,\mathsf{\xi }\in [1,e],\hfill \\ {}^{H}D_{1+}^{11/3}\mathfrak{v}\left(\mathsf{\xi }\right)+\mathfrak{g}(\mathsf{\xi },\mathfrak{u}\left(\mathsf{\xi }\right),\mathfrak{v}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{42/5}\mathfrak{u}\left(\mathsf{\xi }\right){,}^H{I}_{1+}^{29/72}\mathfrak{v}\left(\mathsf{\xi }\right))=0,\mathsf{\xi }\in [1,e],\hfill \end{array}\right.$$

(40)

subject to the boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \mathfrak{u}\left(1\right)={\mathfrak{u}}^{\prime }\left(1\right)=0,{}^{H}D_{1+}^{8/7}\mathfrak{u}\left(e\right)=\frac{1}{6}{}^{H}D_{1+}^{4/9}\mathfrak{u}\left({\displaystyle \frac{3}{2}}\right)+\frac{5}{7}{\int }_2^e{(\nu -1)}^{2/5}{}^{H}D_{1+}^{4/9}\mathfrak{u}\left(\nu \right)d\nu }\hfill \\ {\displaystyle +\frac{1}{3}\mathfrak{v}\left(2\right)+{\int }_{5/2}^e{(\nu -2)}^{1/3}\mathfrak{v}\left(\nu \right)d\nu ,}\hfill \\ {\displaystyle \mathfrak{v}\left(1\right)={\mathfrak{v}}^{\prime }\left(1\right)={\mathfrak{v}}^{\prime \prime }\left(1\right)=0,{}^{H}D_{1+}^{12/5}\mathfrak{v}\left(e\right)=3{\int }_1^{6/5}{\left({\displaystyle \nu -\frac{1}{3}}\right)}^{1/4}{}^{H}D_{1+}^{2/3}\mathfrak{u}\left(\nu \right)d\nu }\hfill \\ {\displaystyle +\frac{1}{2}{}^{H}D_{1+}^{2/3}\mathfrak{u}\left({\displaystyle \frac{15}{7}}\right)+\frac{8}{11}{\int }_1^{4/3}{\left({\displaystyle \nu -\frac{1}{4}}\right)}^{-2/11}{}^{H}D_{1+}^{19/12}\mathfrak{v}\left(\nu \right)d\nu .}\hfill \end{array}\right.$$

(41)

We obtain ${\mathsf{\Xi }}_1\approx 0.47514774$, ${\mathsf{\Xi }}_2\approx 0.29121878$, ${\mathsf{\Xi }}_3\approx 0.71367096$, ${\mathsf{\Xi }}_4\approx 4.31936708$, and $\Delta \approx 1.8445031>0$. So, assumption $\left(I1\right)$ is satisfied. We also find

$${\displaystyle g_1(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(9/4)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{5/4}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{5/4},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{5/4}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

$${\displaystyle g_2(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(11/3)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{8/3}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{8/3},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{8/3}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

$${\displaystyle g_{11}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(65/36)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{29/36}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{29/36},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{29/36}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

$${\displaystyle g_{21}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(19/12)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{7/12}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{7/12},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{7/12}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{3/28},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

$${\displaystyle g_{31}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(11/3)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{8/3}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{8/3},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{8/3}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

$${\displaystyle g_{41}(\mathsf{\xi },\nu )=\frac{1}{\mathsf{\Gamma }(25/12)}\left\{\begin{array}{c}{\left(\ln \mathsf{\xi }\right)}^{13/12}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15}-{\left({\displaystyle \ln \frac{\mathsf{\xi }}{\nu }}\right)}^{13/12},1\le \nu \le \mathsf{\xi }\le e,\hfill \\ {\left(\ln \mathsf{\xi }\right)}^{13/12}{\left({\displaystyle \ln \frac{e}{\nu }}\right)}^{4/15},1\le \mathsf{\xi }\le \nu \le e,\hfill \end{array}\right.}$$

and

$$\begin{array}{c}{\displaystyle {\mathcal{G}}_1(\mathsf{\xi },\nu )=g_1(\mathsf{\xi },\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{5/4}}{\Delta }\left\{{\mathsf{\Xi }}_4\left[{\displaystyle \frac{1}{6}{g}_{11}\left({\displaystyle \frac{3}{2},\nu }\right)+\frac{5}{7}{\int }_2^e{(\rho -1)}^{2/5}{g}_{11}(\rho ,\nu )d\rho }\right]\right.}\hfill \\ +\left.{\mathsf{\Xi }}_2\left[{\displaystyle 3{\int }_1^{6/5}{\left(\rho -\frac{1}{3}\right)}^{1/4}{g}_{21}(\rho ,\nu )d\rho +\frac{1}{2}{g}_{21}\left({\displaystyle \frac{15}{7},\nu }\right)}\right]\right\},\hfill \\ {\displaystyle {\mathcal{G}}_2(\mathsf{\xi },\nu )=\frac{{\left(\ln \mathsf{\xi }\right)}^{5/4}}{\Delta }\left\{{\mathsf{\Xi }}_4\left[{\displaystyle \frac{1}{3}{g}_{31}(2,\nu )+{\int }_{5/2}^e{(\rho -2)}^{1/3}{g}_{31}(\rho ,\nu )d\rho }\right]\right.}\hfill \\ +\left.{\mathsf{\Xi }}_2\left({\displaystyle \frac{8}{11}{\int }_1^{4/3}{\left({\displaystyle \rho -\frac{1}{4}}\right)}^{-2/11}{g}_{41}(\rho ,\nu )d\rho }\right)\right\},\hfill \\ {\displaystyle {\mathcal{G}}_3(\mathsf{\xi },\nu )=\frac{{\left(\ln \mathsf{\xi }\right)}^{8/3}}{\Delta }\left\{{\mathsf{\Xi }}_3\left[{\displaystyle \frac{1}{6}{g}_{11}\left({\displaystyle \frac{3}{2},\nu }\right)+\frac{5}{7}{\int }_2^e{(\rho -1)}^{2/5}{g}_{11}(\rho ,\nu )d\rho }\right]\right.}\hfill \\ +\left.{\mathsf{\Xi }}_1\left[{\displaystyle 3{\int }_1^{6/5}{\left({\displaystyle \rho -\frac{1}{3}}\right)}^{1/4}{g}_{21}(\rho ,\nu )d\rho +\frac{1}{2}{g}_{21}\left({\displaystyle \frac{15}{7},\nu }\right)}\right]\right\},\hfill \\ {\displaystyle {\mathcal{G}}_4(\mathsf{\xi },\nu )=g_2(\mathsf{\xi },\nu )+\frac{{\left(\ln \mathsf{\xi }\right)}^{8/3}}{\Delta }\left\{{\mathsf{\Xi }}_3\left[{\displaystyle \frac{1}{3}{g}_{31}(2,\nu )+{\int }_{5/2}^e{(\rho -2)}^{1/3}{g}_{31}(\rho ,\nu )d\rho }\right]\right.}\hfill \\ +\left.{\mathsf{\Xi }}_1\left({\displaystyle \frac{8}{11}{\int }_1^{4/3}{\left({\displaystyle \rho -\frac{1}{4}}\right)}^{-2/11}{g}_{41}(\rho ,\nu )d\rho }\right)\right\},\hfill \end{array}$$

for all $\mathsf{\xi },\nu \in [1,e]$.

We take $\omega =2$. After some complex computations in Mathematica program, which include computations of some double integrals, we deduce $A_1={\int }_1^e{\mathcal{G}}_1(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 1.12980524$, $A_2={\int }_1^e{\mathcal{G}}_2(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.26094395$, $A_3={\int }_1^e{\mathcal{G}}_3(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.18146763$, $A_4={\int }_1^e{\mathcal{G}}_4(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.14998244$, ${\tilde{A}}_1={\int }_2^e{\mathcal{G}}_1(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.52665008$, ${\tilde{A}}_2={\int }_2^e{\mathcal{G}}_2(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.03066644$, ${\tilde{A}}_3={\int }_2^e{\mathcal{G}}_3(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.09036931$, ${\tilde{A}}_4={\int }_2^e{\mathcal{G}}_4(e,\nu ){\displaystyle \frac{d\nu }{\nu }}\approx 0.04949263$, $C_1\approx 1.31127287$, $C_2\approx 0.41092639$, ${\tilde{C}}_1\approx 0.36709016$, ${\tilde{C}}_2\approx 0.03801933$.

Example 1. 

We consider the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{{\mathsf{\xi }}^{1/2}{e}^{-3\mathsf{\xi }+2}}{4}{sin}^2(\mathrm{u}+1)+\frac{2{(\mathsf{\xi }+3)}^2{e}^{-5\mathsf{\xi }+2}}{5}\sqrt{{\mathrm{v}}^2+4}}\hfill \\ {\displaystyle +\frac{2\ln \mathsf{\xi }}{\mathsf{\xi }+7}\mathrm{x}+\frac{({\mathsf{\xi }}^{3/4}+1)\mathrm{y}}{9({\mathrm{y}}^2+1)}+\frac{{\mathsf{\xi }}^{17/4}}{{\mathsf{\xi }}^2+3},}\hfill \\ {\displaystyle \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{3(\mathsf{\xi }+1)}{7\mathsf{\xi }}arctan\mathrm{u}+\frac{e^{3\mathsf{\xi }}}{4{(\mathsf{\xi }+2)}^5}{cos}^2\mathrm{v}}\hfill \\ {\displaystyle +\frac{{\mathsf{\xi }}^{1/3}-1}{2(\mathsf{\xi }+6)}\sqrt{{\mathrm{x}}^2+1}+\frac{e^{-\mathsf{\xi }}\mathrm{y}}{8(1+\mathrm{y})}+\frac{{(\mathsf{\xi }-1)}^{19/5}}{{\mathsf{\xi }}^2+4},}\hfill \end{array}$$

(42)

for all $\mathsf{\xi }\in [1,e]$ and $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\ge 0$. We obtain the following inequalities

$$\begin{array}{c}{\displaystyle |\mathfrak{f}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)-\mathfrak{f}(\mathsf{\xi },{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3,{\mathrm{y}}_4)|\le \sum _{i=1}^4{X}_i\left(\mathsf{\xi }\right)|{\mathrm{x}}_i-{\mathrm{y}}_i|,}\hfill \\ {\displaystyle |\mathfrak{g}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)-\mathfrak{g}(\mathsf{\xi },{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{y}}_3,{\mathrm{y}}_4)|\le \sum _{i=1}^4{Y}_i\left(\mathsf{\xi }\right)|{\mathrm{x}}_i-{\mathrm{y}}_i|,}\hfill \end{array}$$

for all $\mathsf{\xi }\in [1,e]$ and ${\mathrm{x}}_i,{\mathrm{y}}_i\in {\mathbb{R}}_{+}$, $i=1,\dots ,4$, where

$$\begin{array}{c}{\displaystyle X_1\left(\mathsf{\xi }\right)=\frac{{\mathsf{\xi }}^{1/2}{e}^{-3\mathsf{\xi }+2}}{2},X_2\left(\mathsf{\xi }\right)=\frac{2{(\mathsf{\xi }+3)}^2{e}^{-5\mathsf{\xi }+2}}{5},X_3\left(\mathsf{\xi }\right)=\frac{2\ln \mathsf{\xi }}{\mathsf{\xi }+7},X_4\left(\mathsf{\xi }\right)=\frac{{\mathsf{\xi }}^{3/4}+1}{9},}\hfill \\ {\displaystyle Y_1\left(\mathsf{\xi }\right)=\frac{3(\mathsf{\xi }+1)}{7\mathsf{\xi }},Y_2\left(\mathsf{\xi }\right)=\frac{e^{3\mathsf{\xi }}}{2{(\mathsf{\xi }+2)}^5},Y_3\left(\mathsf{\xi }\right)=\frac{{\mathsf{\xi }}^{1/3}-1}{2(\mathsf{\xi }+6)},Y_4\left(\mathsf{\xi }\right)=\frac{e^{-\mathsf{\xi }}}{8},\mathsf{\xi }\in [1,e].}\hfill \end{array}$$

The functions $X_i,Y_i,i=1,\dots ,4$, are continuous and satisfy the condition (24). In addition, we find $x_1^{*}\approx 0.18393972$, $x_2^{*}\approx 0.31863724$, $x_3^{*}\approx 0.20579769$, $x_4^{*}\approx 0.34633334$, $y_1^{*}\approx 0.85714286$, $y_2^{*}\approx 0.74414061$, $y_3^{*}\approx 0.02268867$, $y_4^{*}\approx 0.04598493$, ${\mathsf{{\rm Y}}}_1\approx 0.89624512$, ${\mathsf{{\rm Y}}}_2\approx 0.75463775$, ${\mathsf{{\rm Y}}}_0={\mathsf{{\rm Y}}}_1<1$. So, assumption (25) is verified. Then, by Theorem 1, we conclude that Problem (40),(41) with the nonlinearities (42) has a unique positive solution $({\mathfrak{u}}^{*}\left(\mathsf{\xi }\right),{\mathfrak{v}}^{*}\left(\mathsf{\xi }\right)),\mathsf{\xi }\in [1,e]$.

Example 2. 

We consider the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{{(2\mathsf{\xi }+3)}^{1/4}{e}^{-5\mathsf{\xi }+2}\mathrm{u}}{7}+\frac{{(\mathsf{\xi }-1)}^3\mathrm{v}}{4(\mathsf{\xi }+6)({\mathrm{v}}^2+2)}}\hfill \\ {\displaystyle +\frac{(\mathsf{\xi }+2)\mathrm{x}}{9(1+2\mathrm{x})}+\frac{e^{\mathsf{\xi }}\mathrm{y}}{5}+\frac{{(\mathsf{\xi }+1)}^{15/4}}{{\mathsf{\xi }}^2+7},}\hfill \\ {\displaystyle \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{{\mathsf{\xi }}^{3/5}\mathrm{u}}{4(\mathsf{\xi }+1)(1+2{\mathrm{u}}^2)}+\frac{(\mathsf{\xi }+2)e^{-\mathsf{\xi }+3}\mathrm{v}}{6({\mathsf{\xi }}^2+5)}}\hfill \\ {\displaystyle +\frac{2\mathsf{\xi }\mathrm{x}}{{\mathsf{\xi }}^3+4}+\frac{\mathrm{y}}{{(\mathsf{\xi }+1)}^3(1+4\mathrm{y})}+\frac{{\mathsf{\xi }}^{8/3}}{2\mathsf{\xi }+5},}\hfill \end{array}$$

(43)

for all $\mathsf{\xi }\in [1,e]$ and $\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\ge 0$. We obtain the inequalities

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)\le \sum _{i=1}^4{U}_i\left(\mathsf{\xi }\right){\mathrm{x}}_i+U_5\left(\mathsf{\xi }\right),\mathfrak{g}(\mathsf{\xi },{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{x}}_4)\le \sum _{i=1}^4{V}_i\left(\mathsf{\xi }\right){\mathrm{x}}_i+V_5\left(\mathsf{\xi }\right),}\hfill \end{array}$$

for all $\mathsf{\xi }\in [1,e]$, ${\mathrm{x}}_i\in {\mathbb{R}}_{+}$, $i=1,\dots ,4$, where

$$\begin{array}{c}{\displaystyle U_1\left(\mathsf{\xi }\right)=\frac{{(2\mathsf{\xi }+3)}^{1/4}{e}^{-5\mathsf{\xi }+2}}{7},U_2\left(\mathsf{\xi }\right)=\frac{{(\mathsf{\xi }-1)}^3}{8(\mathsf{\xi }+6)},U_3\left(\mathsf{\xi }\right)=\frac{\mathsf{\xi }+2}{9},}\hfill \\ {\displaystyle U_4\left(\mathsf{\xi }\right)=\frac{e^{\mathsf{\xi }}}{5},U_5\left(\mathsf{\xi }\right)=\frac{{(\mathsf{\xi }+1)}^{15/4}}{{\mathsf{\xi }}^2+7},}\hfill \\ {\displaystyle V_1\left(\mathsf{\xi }\right)=\frac{{\mathsf{\xi }}^{3/5}}{4(\mathsf{\xi }+1)},V_2\left(\mathsf{\xi }\right)=\frac{(\mathsf{\xi }+2)e^{-\mathsf{\xi }+3}}{6({\mathsf{\xi }}^2+5)},V_3\left(\mathsf{\xi }\right)=\frac{2\mathsf{\xi }}{{\mathsf{\xi }}^3+4},}\hfill \\ {\displaystyle V_4\left(\mathsf{\xi }\right)=\frac{1}{{(\mathsf{\xi }+1)}^3},V_5\left(\mathsf{\xi }\right)=\frac{{\mathsf{\xi }}^{8/3}}{2\mathsf{\xi }+5},\forall \mathsf{\xi }\in [1,e].}\hfill \end{array}$$

The functions $U_i,V_i,i=1,\dots ,5$, are continuous and satisfy the condition (29). In addition, we find $u_1^{*}\approx 0.01063558$, $u_2^{*}\approx 0.07273816$, $u_3^{*}\approx 0.52425354$, $u_4^{*}\approx 3.03085245$, $u_5^{*}\approx 9.56646165$, $v_1^{*}\approx 0.12754245$, $v_2^{*}\approx 0.61575467$, $v_3^{*}\approx 0.41997368$, $v_4^{*}=0.125$, $v_5^{*}\approx 1.37898992$, ${\mathsf{\Lambda }}_1\approx 0.83778721$, ${\mathsf{\Lambda }}_2\approx 0.491456$, ${\mathsf{\Lambda }}_0={\mathsf{\Lambda }}_1<1$. So, Assumption (30) is satisfied. Then, by Theorem 2, we deduce that Problem (40),(41) with the functions (43) has at least one positive solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right),\mathsf{\xi }\in [1,e]$.

Example 3. 

We consider the functions

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{{(\mathrm{u}+\mathrm{v})}^3}{21}+\frac{1}{30}{sin}^2\mathrm{x}+\frac{1}{18}arctan\mathrm{y}+\frac{2({\mathsf{\xi }}^2+3)}{19(\mathsf{\xi }+5)},}\hfill \\ {\displaystyle \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})=\frac{1}{4}{e}^{-2(\mathrm{u}+\mathrm{v})}+\frac{\mathrm{x}}{3}+\frac{{\mathrm{y}}^3}{8}+\frac{3(\mathsf{\xi }+2)}{5({\mathsf{\xi }}^2+6)},}\hfill \end{array}$$

(44)

for all $\mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y}\in {\mathbb{R}}_{+}$. The assumption $\left(I2\right)$ is satisfied. We choose $R_1={\displaystyle \frac{1}{50}}$, $R_2=1$, ${\lambda }_1={\displaystyle \frac{1}{2}}<C_1^{-1}\approx 0.76261778$, ${\lambda }_2=2<C_2^{-1}\approx 2.43352586$, ${\lambda }_3=3>{\tilde{C}}_1^{-1}\approx 2.72412644$, and ${\lambda }_4=30>{\tilde{C}}_2^{-1}\approx 26.30241043$.

Then, we obtain

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\ge \underset{\mathsf{\xi }\in [2,e]}{min}\frac{2({\mathsf{\xi }}^2+3)}{19(\mathsf{\xi }+5)}\approx 0.10526316\ge \frac{{\lambda }_3{R}_1}{2}\approx 0.03,}\hfill \\ {\displaystyle \forall \mathsf{\xi }\in [2,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le \frac{1}{50};\mathrm{x}\in \left[{\displaystyle 0,\frac{1}{50\mathsf{\Gamma }(41/26)}}\right],\mathrm{y}\in \left[{\displaystyle 0,\frac{1}{50\mathsf{\Gamma }(38/7)}}\right],}\hfill \\ {\displaystyle \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\ge \frac{1}{4}{e}^{-1/25}+\underset{\mathsf{\xi }\in [2,e]}{min}\frac{3(\mathsf{\xi }+2)}{5({\mathsf{\xi }}^2+6)}\approx 0.45163639\ge \frac{{\lambda }_4{R}_1}{2}=0.3,}\hfill \\ {\displaystyle \forall \mathsf{\xi }\in [2,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le \frac{1}{50},\mathrm{x}\in \left[{\displaystyle 0,\frac{1}{50\mathsf{\Gamma }(47/5)}}\right],\mathrm{y}\in \left[{\displaystyle 0,\frac{1}{50\mathsf{\Gamma }(101/72)}}\right].}\hfill \end{array}$$

So, assumption $\left(I3\right)$ is satisfied. In addition, we find

$$\begin{array}{c}{\displaystyle \mathfrak{f}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le \frac{1}{21}+\frac{1}{30}+\frac{1}{18}arctan\frac{1}{\mathsf{\Gamma }(38/7)}+\underset{\mathsf{\xi }\in [1,e]}{max}\frac{2({\mathsf{\xi }}^2+3)}{19(\mathsf{\xi }+5)}\approx 0.22383002}\hfill \\ {\displaystyle \le \frac{{\lambda }_1{R}_2}{2}=0.25,\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le 1,\mathrm{x}\in \left[{\displaystyle 0,\frac{1}{\mathsf{\Gamma }(41/26)}}\right],\mathrm{y}\in \left[{\displaystyle 0,\frac{1}{\mathsf{\Gamma }(38/7)}}\right],}\hfill \\ {\displaystyle \mathfrak{g}(\mathsf{\xi },\mathrm{u},\mathrm{v},\mathrm{x},\mathrm{y})\le \frac{1}{4}+\frac{1}{3\mathsf{\Gamma }(47/5)}+\frac{1}{8}{\left({\displaystyle \frac{1}{\mathsf{\Gamma }(101/72)}}\right)}^3+\underset{\mathsf{\xi }\in [1,e]}{max}\frac{3(\mathsf{\xi }+2)}{5({\mathsf{\xi }}^2+6)}\approx 0.68716505}\hfill \\ {\displaystyle \le \frac{{\lambda }_2{R}_2}{2}=1,\forall \mathsf{\xi }\in [1,e],\mathrm{u},\mathrm{v}\ge 0,\mathrm{u}+\mathrm{v}\le 1,\mathrm{x}\in \left[0,\frac{1}{\mathsf{\Gamma }(47/5)}\right],\mathrm{y}\in \left[0,\frac{1}{\mathsf{\Gamma }(101/72)}\right].}\hfill \end{array}$$

So, assumption $\left(I4\right)$ is also satisfied. By Theorem 3, we conclude that Problem (40),(41) with the nonlinearities (44) has at least one positive solution $\left(\mathfrak{u}\right(\mathsf{\xi }),\mathfrak{v}(\mathsf{\xi }\left)\right),\mathsf{\xi }\in [1,e]$, such that ${\displaystyle \frac{1}{50}}\le \parallel \mathfrak{u}\parallel +\parallel \mathfrak{v}\parallel \le 1$ and $\mathfrak{u}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{5/4}\parallel \mathfrak{u}\parallel$, $\mathfrak{v}\left(\mathsf{\xi }\right)\ge {\left(\ln \mathsf{\xi }\right)}^{8/3}\parallel \mathfrak{v}\parallel$ for all $\mathsf{\xi }\in [1,e]$.

5. Conclusions

Our paper investigates the positive solutions of the system of Hadamard fractional differential equations (1), focusing on their existence, uniqueness, and multiplicity. These equations include fractional integral terms and are subject to boundary conditions (2) that involve various Hadamard fractional derivatives and Riemann–Stieltjes integrals. These coupled boundary conditions are general and involve the dependence of the fractional derivatives of u and v at the point e on different derivatives of both functions u and v. In our main findings, we employed several fixed-point theorems, such as the Banach contraction mapping principle, the Schauder fixed-point theorem, the Guo–Krasnosel’skii fixed-point theorem, and the Leggett–Williams fixed-point theorem.