On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

Abstract

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

1. Introduction

We shall examine a system comprising nonlinear Hadamard fractional differential equations

$$\left\{\begin{array}{c}{}^H{D}_{1+}^{\alpha }\Phi \left(\upsilon \right)+\mathfrak{c}\left(\upsilon \right)\mathfrak{L}(\Phi \left(\upsilon \right),\Psi \left(\upsilon \right))=0,\upsilon \in (1,\infty ),\hfill \\ {}^H{D}_{1+}^{\beta }\Psi \left(\upsilon \right)+\mathfrak{d}\left(\upsilon \right)\mathfrak{M}(\Phi \left(\upsilon \right),\Psi \left(\upsilon \right))=0,\upsilon \in (1,\infty ),\hfill \end{array}\right.$$

(1)

supplemented with the nonlocal coupled boundary conditions

$$\left\{\begin{array}{c}\Phi \left(1\right)=\Phi \prime \left(1\right)=\cdots ={\Phi }^{(n-2)}\left(1\right)=0,\hfill \\ {\displaystyle {}^H{D}_{1+}^{\alpha -1}\Phi (\infty )={\int }_1^{\infty }\Phi \left(\omega \right)d{\mathfrak{I}}_1\left(\omega \right)+{\int }_1^{\infty }\Psi \left(\omega \right)d{\mathfrak{I}}_2\left(\omega \right),}\hfill \\ \Psi \left(1\right)=\Psi \prime \left(1\right)=\cdots ={\Psi }^{(m-2)}\left(1\right)=0,\hfill \\ {\displaystyle {}^H{D}_{1+}^{\beta -1}\Psi (\infty )={\int }_1^{\infty }\Phi \left(\omega \right)d{\mathfrak{J}}_1\left(\omega \right)+{\int }_1^{\infty }\Psi \left(\omega \right)d{\mathfrak{J}}_2\left(\omega \right),}\hfill \end{array}\right.$$

(2)

where $\alpha \in (n-1,n]$, $n\in \mathbb{N}$, $n\ge 2$, $\beta \in (m-1,m]$, $m\in \mathbb{N}$, $m\ge 2$, ${}^H{D}_{1+}^{\kappa }$ denotes the Hadamard fractional derivative of order p (for $p=\alpha , \alpha -1, \beta , \beta -1$), the functions $\mathfrak{c},\mathfrak{d}:(1,\infty )\to {\mathbb{R}}_{+}$ and $\mathfrak{L},\mathfrak{M}:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ verify some assumptions, (${\mathbb{R}}_{+}=[0,\infty )$), and the functions ${\mathfrak{I}}_i,{\mathfrak{J}}_i:[1,\infty )\to \mathbb{R}$, $i=1, 2$ from the Riemann–Stieltjes integrals of (2) have bounded variations.

By employing the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem, we will establish the existence of positive solutions of the problem (1),(2) subject to certain conditions on the problem’s data. A positive solution of (1),(2) is represented by a pair of functions $(\Phi (\upsilon ),\Psi (\upsilon \left)\right),\upsilon \in [1,\infty )$ satisfying (1) and (2), with $\Phi \left(\zeta \right)\ge 0$, $\Psi \left(\zeta \right)\ge 0$ for all $\zeta \in [1,\infty )$, and $\Phi \left(\zeta \right)>0$ for all $\zeta \in (1,\infty )$ or $\Psi \left(\zeta \right)>0$ for all $\zeta \in (1,\infty )$. This problem generalizes the problem from [1]. In the paper [1], the authors studied the system (1) with $\alpha ,\beta \in (1,2]$ ($n=m=2$), with the boundary conditions

$$\left\{\begin{array}{c}{\displaystyle \Phi \left(1\right)=0,{}^H{D}_{1+}^{\alpha -1}\Phi (\infty )=\sum _{i=1}^m{\lambda }_i{}^H{I}_{1+}^{{\alpha }_i}\Psi \left(\eta \right),}\hfill \\ {\displaystyle \Psi \left(1\right)=0,{}^H{D}_{1+}^{\beta -1}\Psi (\infty )=\sum _{j=1}^n{\sigma }_j{}^H{I}_{1+}^{{\beta }_j}\Phi \left(\xi \right),}\hfill \end{array}\right.$$

(3)

where ${\lambda }_i,{\sigma }_j>0$ for $i=1, \dots , m$, $j=1, \dots , n$, $\eta >1$, $\xi >1$, and ${}^H{I}_{1+}^k$ is the Hadamard fractional integral of order k with lower limit 1 for $k={\alpha }_1, \dots , {\alpha }_m, {\beta }_1, \dots , {\beta }_n$, defined by

$${\displaystyle {}^H{I}_{1+}^k\mathrm{h}\left(\upsilon \right)=\frac{1}{\Gamma \left(k\right)}{\int }_1^{\upsilon }{\left(ln\frac{\upsilon }{\omega }\right)}^{k-1}\frac{\mathrm{h}\left(\omega \right)}{\omega }d\omega ,\upsilon >0,}$$

(for a function $\mathrm{h}:[1,\infty )\to \mathbb{R}$), (see [2]). The last conditions for ∞ from (3) are particular cases of our conditions from (2). Indeed, one finds

$${\displaystyle \sum _{i=1}^m{\lambda }_i{}^H{I}_{1+}^{{\alpha }_i}\Psi \left(\eta \right)={\int }_1^{\infty }\Psi \left(\omega \right)d{\mathfrak{I}}_2\left(\omega \right),\mathrm{and}\sum _{j=1}^n{\sigma }_j{}^H{I}_{1+}^{{\beta }_j}\Phi \left(\xi \right)={\int }_1^{\infty }\Phi \left(\omega \right)d{\mathfrak{J}}_1\left(\omega \right),}$$

with ${\mathfrak{I}}_2\left(\omega \right)={\sum }_{i=1}^m{\lambda }_i{\tilde{H}}_i\left(\omega \right)$, (${\mathfrak{I}}_1\equiv 0$), and ${\mathfrak{J}}_1\left(\omega \right)={\sum }_{j=1}^n{\sigma }_j{\tilde{K}}_j\left(\omega \right)$, (${\mathfrak{J}}_2\equiv 0$), where

$${\tilde{H}}_i\left(\omega \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma ({\alpha }_i+1)}\left({(ln\eta )}^{{\alpha }_i}-{\left(ln\frac{\eta }{\omega }\right)}^{{\alpha }_i}\right),\mathrm{if}0\le \omega \le \eta ,}\hfill \\ {\displaystyle \frac{1}{\Gamma ({\alpha }_i+1)}{(ln\eta )}^{{\alpha }_i},\mathrm{if}\omega \ge \eta ,}\hfill \end{array}\right.$$

and

$${\tilde{K}}_j\left(\omega \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma ({\beta }_j+1)}\left({(ln\xi )}^{{\beta }_j}-{\left(ln\frac{\xi }{\omega }\right)}^{{\beta }_j}\right),\mathrm{if}0\le \omega \le \xi ,}\hfill \\ {\displaystyle \frac{1}{\Gamma ({\beta }_j+1)}{(ln\xi )}^{{\beta }_j},\mathrm{if}\omega \ge \xi ,}\hfill \end{array}\right.$$

for $i=1, \dots , m$, and $j=1, \dots , n$. Therefore, in our paper, we consider general orders for the fractional derivatives in the equations of system (1). Furthermore, in the boundary conditions (2), the fractional derivatives ${}^H{D}_{1+}^{\alpha -1}\Phi$ and ${}^H{D}_{1+}^{\beta -1}\Psi$ for ∞ are dependent on both functions $\Phi$ and $\Psi$, in comparison with the condition (3) from [1], where the derivative of $\Phi$ is dependent only on $\Psi$, and the derivative of $\Psi$ is dependent only on $\Phi$. In addition, the conditions (2) with Riemann–Stieltjes integrals generalize, as we saw before, the conditions (3). These aspects represent the novelties for our problem (1),(2).

In the upcoming discussion, we will introduce additional fractional boundary value problems that have been investigated in recent years. These problems are connected to the main problem (1),(2) and involve Hadamard fractional differential equations accompanied by diverse nonlocal boundary conditions. In [3], the authors studied the existence of non-negative multiple solutions for the Hadamard fractional differential equation with integro-differential boundary conditions

$$\left\{\begin{array}{c}{}^H{D}_{1+}^{\alpha }\mathfrak{U}\left(\upsilon \right)+\mathrm{a}\left(\upsilon \right)\mathrm{f}\left(\mathfrak{U}\left(\upsilon \right)\right)=0,\upsilon \in (0,\infty ),\hfill \\ {\displaystyle \mathfrak{U}\left(1\right)=0,{}^H{D}_{1+}^{\alpha -1}\mathfrak{U}(\infty )=\sum _{i=1}^m{\lambda }_i{}^H{I}_{1+}^{{\beta }_i}\mathfrak{U}\left(\eta \right),}\hfill \end{array}\right.$$

where $\alpha \in (1,2]$, $\eta \in (1,\infty )$, ${\beta }_i>0$ and ${\lambda }_i\ge 0$ for all $i=1, \dots , m$. In the proofs of the main results of [3], they applied the Leggett–Williams and Guo–Krasnosel’skii fixed point theorems. In [4], by using the Banach fixed point theorem, the authors investigated the existence of the unique solution for the Hadamard fractional integro-differential equation subject to Hadamard fractional integral boundary conditions

$$\left\{\begin{array}{c}{}^H{D}_{1+}^{\gamma }\mathfrak{U}\left(\upsilon \right)+\mathrm{f}(\upsilon ,\mathfrak{U}\left(\upsilon \right){,}^H{I}_{1+}^q\mathfrak{U}\left(\upsilon \right))=0,\upsilon \in (1,\infty ),\hfill \\ {\displaystyle \mathfrak{U}\left(1\right)=\mathfrak{U}\prime \left(1\right)=0,{}^H{D}_{1+}^{\gamma -1}\mathfrak{U}(\infty )=\sum _{i=1}^m{\lambda }_i{}^H{I}_{1+}^{{\beta }_i}\mathfrak{U}\left(\eta \right),}\hfill \end{array}\right.$$

where $\gamma \in (2,3)$, $\eta \in (1,\infty )$, $q>0$, ${\beta }_i>0$ and ${\lambda }_i\ge 0$ for all $i=1, \dots , m$. In [5], the authors studied the existence of positive solutions for the Hadamard fractional differential equation supplemented with Hadamard integral and multi-point boundary conditions

$$\left\{\begin{array}{c}{}^H{D}_{1+}^q\Phi \left(\upsilon \right)+\sigma \left(\upsilon \right)\mathrm{f}(\upsilon ,\Phi \left(\upsilon \right))=0,\upsilon \in (1,\infty ),\hfill \\ {\displaystyle \Phi \left(1\right)=\Phi \prime \left(1\right)=0,{}^H{D}_{1+}^{q-1}\Phi (\infty )=\mathrm{a}{}^H{I}_{1+}^{\beta }\Phi \left(\xi \right)+\mathrm{b}\sum _{j=1}^{m-2}{\alpha }_j\Phi \left({\eta }_j\right),}\hfill \end{array}\right.$$

where $q\in (2,3]$, $1<\xi <{\eta }_1<\cdots <{\eta }_{m-2}<\infty$, $\mathrm{a},\mathrm{b}\in \mathbb{R}$, and ${\alpha }_i\ge 0$ for all $i=1, \dots , m-2$. In the main theorems, they used the monotone iterative method to find two “twin” positive solutions of the problem, and then presented monotone iterative schemes convergent to a unique positive solution. In [6], the authors investigated the nonlinear Hadamard fractional differential equation with nonlocal boundary conditions

$$\left\{\begin{array}{c}{}^H{D}_{1+}^{\alpha }\Phi \left(\upsilon \right)+\mathfrak{c}\left(\upsilon \right)\mathfrak{L}(\upsilon ,\Phi \left(\upsilon \right))=0,\upsilon \in (1,\infty ),\hfill \\ {\displaystyle \Phi \left(1\right)=\Phi \prime \left(1\right)=0,{}^H{D}_{1+}^{\alpha -1}\Phi (\infty )=\sum _{i=1}^m{\alpha }_i{}^H{I}_{1+}^{{\beta }_i}\Phi \left(\eta \right)+b\sum _{j=1}^n{\sigma }_j\Phi \left({\xi }_j\right),}\hfill \end{array}\right.$$

(4)

with $\alpha \in (2,3)$, ${\beta }_i>0$ and ${\alpha }_i\ge 0$ for all $i=1, \dots , m$, ${\sigma }_j\ge 0$ for all $j=1, \dots , n$, $1<\eta <{\xi }_1<\dots <{\xi }_n<\infty$. They studied the existence, uniqueness and multiplicity of positive solutions for problem (4), by applying the Schauder fixed point theorem, the Banach fixed point theorem, the monotone iterative method, and a fixed point theorem due to Avery and Peterson. In [7], the authors proved a generalization of a fixed point theorem due to Avery and Henderson, and then they applied it to problem (4) and proved the existence of at least three positive solutions of (4). To explore further developments in the realm of Hadamard fractional differential equations, we encourage readers to refer to the following papers: [8,9,10,11,12,13,14,15]. Additionally, we would like to highlight the significance of the monographs [2,16,17,18,19,20,21,22,23,24,25], which delve into various aspects of fractional differential equations and systems, encompassing diverse boundary conditions and their wide-ranging applications across different fields.

The paper is structured as follows: Section 2 presents a collection of preliminary results that are crucial for the subsequent sections. This includes the solution to the corresponding linear problem, an examination of the properties of Green functions, and other relevant aspects. Section 3 focuses on the main existence theorems for the problem (1),(2). Section 4 provides a detailed analysis of an example that serves to illustrate the obtained results. Finally, in Section 5, we draw conclusions based on the findings presented in our paper.

2. Auxiliary Results

Some preliminary results that will be used in our further considerations are presented in this section.

 Definition 1 

([2]). The Hadamard integral of fractional order $p>0$ with lower limit $\mathrm{a}$, ($\mathrm{a}\ge 0$) of a function $\Xi :[\mathrm{a},\infty )\to \mathbb{R}$ is defined by

$${\displaystyle {(}^H{I}_{\mathrm{a}+}^p\Xi )\left(\varphi \right)=\frac{1}{\Gamma \left(p\right)}{\int }_{\mathrm{a}}^{\varphi }{\left({\displaystyle ln\frac{\varphi }{\upsilon }}\right)}^{p-1}\frac{\Xi \left(\upsilon \right)}{\upsilon }d\upsilon ,\varphi >0.}$$

 Definition 2 

([2]). The Hadamard fractional derivative of order $p\ge 0$ with lower limit $\mathrm{a}$, ($\mathrm{a}\ge 0$) of a function $\Xi :[\mathrm{a},\infty )\to \mathbb{R}$ is defined by

$${\displaystyle {}^H{D}_{\mathrm{a}+}^p\Xi \left(\varphi \right)={\left({\displaystyle \varphi \frac{d}{d\varphi }}\right)}^n{}^H{I}_{a+}^{n-p}\Xi \left(\varphi \right)=\frac{1}{\Gamma (n-p)}{\left({\displaystyle \varphi \frac{d}{d\varphi }}\right)}^n{\int }_{\mathrm{a}}^{\varphi }{\left({\displaystyle ln\frac{\varphi }{\upsilon }}\right)}^{n-p-1}\frac{\Xi \left(\upsilon \right)}{\upsilon }d\upsilon ,}$$

with $n=\left[p\right]+1$.

For $p=m\in \mathbb{N}$, then ${}^H{D}_{\mathrm{a}+}^m\Xi \left(\varphi \right)=({\delta }^m\Xi )\left(\varphi \right),\varphi >\mathrm{a}$, where $\delta =\varphi \frac{d}{d\varphi }$ is the $\delta$-derivative.

 Lemma 1 

([2]). If $\alpha ,\beta >0$, and $\mathrm{a}>0$, then

$$\begin{array}{c}{\displaystyle \left({}^H{I}_{\mathrm{a}+}^{\alpha }{\left({\displaystyle ln\frac{\upsilon }{\mathrm{a}}}\right)}^{\beta -1}\right)\left(\varphi \right)=\frac{\Gamma \left(\beta \right)}{\Gamma (\beta +\alpha )}{\left({\displaystyle ln\frac{\varphi }{\mathrm{a}}}\right)}^{\beta +\alpha -1},}\hfill \\ {\displaystyle \left({}^H{D}_{\mathrm{a}+}^{\alpha }{\left({\displaystyle ln\frac{\upsilon }{\mathrm{a}}}\right)}^{\beta -1}\right)\left(\varphi \right)=\frac{\Gamma \left(\beta \right)}{\Gamma (\beta -\alpha )}{\left({\displaystyle ln\frac{\varphi }{\mathrm{a}}}\right)}^{\beta -\alpha -1},}\hfill \end{array}$$

and in particular, ${}^H{D}_{\mathrm{a}+}^p{(ln\frac{\upsilon }{\mathrm{a}})}^{p-j}\left(\varphi \right)=0$, for $j=1, \dots , \left[p\right]+1$.

 Lemma 2 

([2]). Let $p>0$ and $\Xi ,\mathfrak{U}\in C[1,\infty )\cap L^1(0,\infty )$. The solutions of the Hadamard fractional differential equation ${}^H{D}_{1+}^p\Xi \left(\upsilon \right)=0$ are

$${\displaystyle \Xi \left(\upsilon \right)=\sum _{i=1}^n{c}_i{(ln\upsilon )}^{p-i},}$$

and the next formula is satisfied

$${\displaystyle {}^H{I}_{1+}^p{}^H{D}_{1+}^p\mathfrak{U}\left(\upsilon \right)=\mathfrak{U}\left(\upsilon \right)+\sum _{i=1}^n{d}_i{(ln\upsilon )}^{p-i},}$$

with $c_i,d_i\in \mathbb{R}$, $i=1, \dots , n$ and $n=\left[p\right]+1$.

Let us now examine the system comprising fractional differential equations

$$\left\{\begin{array}{c}{}^H{D}_{1+}^{\alpha }\Phi \left(\upsilon \right)+\mathrm{h}\left(\upsilon \right)=0,\upsilon \in (1,\infty ),\hfill \\ {}^H{D}_{1+}^{\beta }\Psi \left(\upsilon \right)+\mathrm{k}\left(\upsilon \right)=0,\upsilon \in (1,\infty ),\hfill \end{array}\right.$$

(5)

where $\mathrm{h},\mathrm{k}\in C([1,\infty ),{\mathbb{R}}_{+})$, with the boundary conditions (2). Let us define the constants:

$$\begin{array}{c}{\displaystyle a=\Gamma \left(\alpha \right)-{\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right),b={\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right),}\hfill \\ {\displaystyle c={\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right),d=\Gamma \left(\beta \right)-{\int }_1^{\infty }{(ln\zeta )}^{\beta -1}d{\mathfrak{J}}_2\left(\zeta \right),}\hfill \\ \Delta =ad-bc.\hfill \end{array}$$

(6)

Lemma 3.

Assume that $a,b,c,d\in \mathbb{R}$ and the functions $\mathrm{h},\mathrm{k}\in C([1,\infty ),{\mathbb{R}}_{+})$ satisfy the conditions ${\int }_1^{\infty }\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }<\infty$ and ${\int }_1^{\infty }\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }<\infty$. If $\Delta \ne 0$, then the solution of problem (5),(2) is given by

$$\begin{array}{c}{\displaystyle \Phi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left[{\displaystyle d{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\hfill \\ {\displaystyle +b{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\right],\upsilon \in [1,\infty ),\hfill \\ {\displaystyle \Psi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left[{\displaystyle a{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{a}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{a}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\hfill \\ {\displaystyle +c{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{c}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{c}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\right],\upsilon \in [1,\infty ).\hfill \end{array}$$

(7)

Proof. 

By using Lemma 2, the solutions of system (5) are given by

$$\begin{array}{c}{\displaystyle \Phi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ +a_1{(ln\upsilon )}^{\alpha -1}+a_2{(ln\upsilon )}^{\alpha -2}+\cdots +a_n{(ln\upsilon )}^{\alpha -n},\upsilon \in [1,\infty ),\hfill \\ {\displaystyle \Psi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ +b_1{(ln\upsilon )}^{\beta -1}+b_2{(ln\upsilon )}^{\beta -2}+\cdots +b_m{(ln\upsilon )}^{\beta -m},\upsilon \in [1,\infty ),\hfill \end{array}$$

(8)

with $a_j,b_k\in \mathbb{R}$ for $j=1, \dots , n$, $k=1, \dots , m$. Because $\Phi \left(1\right)=\Phi \prime \left(1\right)=\cdots ={\Phi }^{(n-2)}\left(1\right)=0$ and $\Psi \left(1\right)=\Psi \prime \left(1\right)=\cdots ={\Psi }^{(m-2)}\left(1\right)=0$, we deduce that $a_j=0$ for $j=2, \dots , n$ and $b_k=0$ for $k=2, \dots , m$. So, by (8), one finds

$$\begin{array}{c}{\displaystyle \Phi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+a_1{(ln\upsilon )}^{\alpha -1},\upsilon \in [1,\infty ),}\hfill \\ {\displaystyle \Psi \left(\upsilon \right)=-\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+b_1{(ln\upsilon )}^{\beta -1},\upsilon \in [1,\infty ).}\hfill \end{array}$$

(9)

We have ${}^H{D}_{1+}^{\alpha -1}\Phi \left(\upsilon \right)=-{\int }_1^{\upsilon }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+a_1\Gamma \left(\alpha \right)$ and ${}^H{D}_{1+}^{\beta -1}\Psi \left(\upsilon \right)=-{\int }_1^{\upsilon }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+b_1\Gamma \left(\beta \right)$. Then, the last conditions from (2), namely ${}^H{D}_{1+}^{\alpha -1}\Phi (\infty )={\int }_1^{\infty }\Phi \left(\omega \right)d{\mathfrak{I}}_1\left(\omega \right)+$${\int }_1^{\infty }\Psi \left(\omega \right)d{\mathfrak{I}}_2\left(\omega \right)$ and ${}^H{D}_{1+}^{\beta -1}\Psi (\infty )={\int }_1^{\infty }\Phi \left(\omega \right)d{\mathfrak{J}}_1\left(\omega \right)+{\int }_1^{\infty }\Psi \left(\omega \right)d{\mathfrak{J}}_2\left(\omega \right)$, for the solution (9), give us

$$\left\{\begin{array}{c}{\displaystyle -{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+a_1\Gamma \left(\alpha \right)={\int }_1^{\infty }\left[{\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }+a_1{(ln\omega )}^{\alpha -1}}\right]d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left[{\displaystyle -\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }+b_1{(ln\omega )}^{\beta -1}}\right]d{\mathfrak{I}}_2\left(\omega \right),}\hfill \\ {\displaystyle -{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+b_1\Gamma \left(\beta \right)={\int }_1^{\infty }\left[{\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }+a_1{(ln\omega )}^{\alpha -1}}\right]d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left[{\displaystyle -\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }+b_1{(ln\omega )}^{\beta -1}}\right]d{\mathfrak{J}}_2\left(\omega \right),}\hfill \end{array}\right.$$

or

$$\left\{\begin{array}{c}{\displaystyle a_1\left[{\displaystyle \Gamma \left(\alpha \right)-{\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right]-b_1{\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\hfill \\ {\displaystyle ={\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{s}{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right),}\hfill \\ {\displaystyle -a_1{\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)+b_1\left[{\displaystyle \Gamma \left(\beta \right)-{\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\right]}\hfill \\ {\displaystyle ={\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right).}\hfill \end{array}\right.$$

By assuming the non-zero determinant $\Delta$ of the aforementioned system (with respect to the unknowns $a_1$ and $b_1$), we establish the uniqueness of the solution for the system as follows:

$$\begin{array}{c}{\displaystyle a_1=\frac{1}{\Delta }\left[{\displaystyle d{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left(ln\frac{\omega }{\xi }\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\hfill \\ {\displaystyle +b{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ \left.{\displaystyle -\frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\right],\hfill \\ {\displaystyle b_1=\frac{1}{\Delta }\left[{\displaystyle a{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{a}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{a}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\hfill \\ {\displaystyle +c{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{c}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{c}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\right].\hfill \end{array}$$

By replacing the above formulas for $a_1$ and $b_1$ in (9), we obtain the solution of problem (5),(2) given by (7). The converse follows by direct computation. □

Lemma 4.

Given the conditions stated in Lemma 3, the solution to problem (5),(2) is determined as follows:

$$\left\{\begin{array}{c}{\displaystyle \Phi \left(\upsilon \right)={\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega },\upsilon \in [1,\infty ),}\hfill \\ {\displaystyle \Psi \left(\upsilon \right)={\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+{\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega },\upsilon \in [1,\infty ),}\hfill \end{array}\right.$$

(10)

with the Green functions ${\mathfrak{E}}_i,i=1, \dots , 4$ given by

$$\begin{array}{c}{\displaystyle {\mathfrak{E}}_1(\upsilon ,\omega )=g_{\alpha }(\upsilon ,\omega )+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left({\displaystyle d{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)+b{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right),}\hfill \\ {\displaystyle {\mathfrak{E}}_2(\upsilon ,\omega )=\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left({\displaystyle d{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)+b{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right),}\hfill \\ {\displaystyle {\mathfrak{E}}_3(\upsilon ,\omega )=\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left({\displaystyle c{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)+a{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right),}\hfill \\ {\displaystyle {\mathfrak{E}}_4(\upsilon ,\omega )=g_{\beta }(\upsilon ,\omega )+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left({\displaystyle c{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)+a{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right),}\hfill \end{array}$$

(11)

and

$$\begin{array}{c}{\displaystyle g_{\alpha }(\upsilon ,\omega )=\frac{1}{\Gamma \left(\alpha \right)}\left\{\begin{array}{c}{(ln\upsilon )}^{\alpha -1}-{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1},1\le \omega \le \upsilon ,\hfill \\ {(ln\upsilon )}^{\alpha -1},1\le \upsilon \le \omega ,\hfill \end{array}\right.}\hfill \\ {\displaystyle g_{\beta }(\upsilon ,\omega )=\frac{1}{\Gamma \left(\beta \right)}\left\{\begin{array}{c}{(ln\upsilon )}^{\beta -1}-{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1},1\le \omega \le \upsilon ,\hfill \\ {(ln\upsilon )}^{\beta -1},1\le \upsilon \le \omega .\hfill \end{array}\right.}\hfill \end{array}$$

(12)

Proof. 

By (7), one obtains

$$\begin{array}{c}{\displaystyle \Phi \left(\upsilon \right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }{(ln\upsilon )}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{\Delta }{\Gamma \left(\alpha \right)\Delta }{\int }_1^{\infty }{(ln\upsilon )}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left[{\displaystyle d{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\hfill \\ {\displaystyle +b{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\right]\hfill \\ =\underset{A_0}{\underbrace{{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{\upsilon }\left[{(ln\upsilon )}^{\alpha -1}-{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\alpha -1}\right]\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{\upsilon }^{\infty }{(ln\upsilon )}^{\alpha -1}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}}}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left[{\displaystyle -\frac{ad-bc}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+d{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\right.}\hfill \\ {\displaystyle -\frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ {\displaystyle -\frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)+b{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\right]\hfill \\ {\displaystyle =A_0+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left\{{\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left[{\displaystyle \Gamma \left(\alpha \right)\Gamma \left(\beta \right)-\Gamma \left(\beta \right){\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right.}\right.}\hfill \\ {\displaystyle -\Gamma \left(\alpha \right){\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)+\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)}\right)}\hfill \\ {\displaystyle -\left.\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}{\mathfrak{I}}_2\left(\omega \right)}\right)\right]{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\left[{\displaystyle \Gamma \left(\beta \right)-{\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)}\right]{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left[{\displaystyle \Gamma \left(\beta \right)-{\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)}\right]{\int }_1^{\infty }\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}\left[{\displaystyle \Gamma \left(\beta \right)-{\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)}\right]{\int }_1^{\infty }\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\right){\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\right)}\hfill \\ -\left.{\displaystyle \frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right)}\right\}\hfill \\ {\displaystyle =A_0+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left[{\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right.}\right.}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle -\frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha \right)}{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{s}}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\hfill \\ {\displaystyle -\left.{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\left({\displaystyle {\int }_{\omega }^{\infty }{\left(ln\frac{\xi }{\omega }\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)}\right]\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left[{\displaystyle -{\int }_{\omega }^{\infty }{\left(ln\frac{\xi }{\omega }\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right.}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\hfill \\ +\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\hfill \\ -\left.{\displaystyle \left.{\displaystyle \frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right]\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle =A_0+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left\{{\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right]}\right.}\right.}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right]}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle \times \left.\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right]\right\}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left\{\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right]\right.}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\beta \right)}\left[\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\right.}\hfill \\ -\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\hfill \\ +\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\hfill \\ -\left.{\displaystyle \left.\left.\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\right]\right\}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle =A_0+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left\{{\displaystyle \frac{d}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right]}\right.}\right.}\hfill \\ {\displaystyle +\left.{\displaystyle \frac{b}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right]}\right\}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left\{{\displaystyle \frac{d}{\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left(ln\frac{\xi }{\omega }\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right]}\right.}\hfill \\ +\left.{\displaystyle \left.{\displaystyle \frac{b}{\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right]}\right\}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle ={\int }_1^{\infty }{g}_{\alpha }(\upsilon ,\omega )\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{(ln\upsilon )}^{\alpha -1}}{\Delta }\left[{\displaystyle d{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\right.}\hfill \\ {\displaystyle +b{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +d{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ +\left.{\displaystyle b{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right]\hfill \\ {\displaystyle ={\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega },t\in [1,\infty ),}\hfill \end{array}$$

where $g_{\alpha }$ and $g_{\beta }$ are given by (12), and ${\mathfrak{E}}_1$ and ${\mathfrak{E}}_2$ are given by (11). In the last relations above, we used the equality

$$\begin{array}{c}{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left(ln\frac{\xi }{\omega }\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\hfill \\ {\displaystyle ={\int }_1^{\omega }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)+{\int }_{\omega }^{\infty }\left[{(ln\xi )}^{\alpha -1}-{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}\right]d{\mathfrak{I}}_1\left(\xi \right)}\hfill \\ {\displaystyle =\Gamma \left(\alpha \right){\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right),}\hfill \end{array}$$

and similar equalities for ${\mathfrak{I}}_2\left(\xi \right),{\mathfrak{J}}_1\left(\xi \right)$ and ${\mathfrak{J}}_2\left(\xi \right)$.

In a similar manner, by (7), one finds

$$\begin{array}{c}{\displaystyle \Psi \left(\upsilon \right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\infty }{(ln\upsilon )}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{\Delta }{\Gamma \left(\beta \right)\Delta }{\int }_1^{\infty }{(ln\upsilon )}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\upsilon }{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left[{\displaystyle a{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }-\frac{a}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\right.}\hfill \\ {\displaystyle -\frac{a}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\hfill \\ {\displaystyle +c{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{c}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{c}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\right]\hfill \\ =\underset{B_0}{\underbrace{{\displaystyle \frac{1}{\Gamma \left(\beta \right)}{\int }_1^{\upsilon }\left[{(ln\upsilon )}^{\beta -1}-{\left({\displaystyle ln\frac{\upsilon }{\omega }}\right)}^{\beta -1}\right]\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+\frac{1}{\Gamma \left(\beta \right)}{\int }_{\upsilon }^{\infty }{(ln\upsilon )}^{\beta -1}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}}}\hfill \\ {\displaystyle +\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left[{\displaystyle -\frac{ad-bc}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+a{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right.}\hfill \\ {\displaystyle -\frac{a}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ {\displaystyle -\frac{a}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\hfill \\ {\displaystyle +c{\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }-\frac{c}{\Gamma \left(\alpha \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\hfill \\ -\left.{\displaystyle \frac{c}{\Gamma \left(\beta \right)}{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\right]\hfill \\ {\displaystyle =B_0+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left\{{\displaystyle -\frac{1}{\Gamma \left(\beta \right)}\left[{\displaystyle \Gamma \left(\alpha \right)\Gamma \left(\beta \right)-\Gamma \left(\beta \right){\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right.}\right.}\hfill \\ {\displaystyle -\Gamma \left(\alpha \right){\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)+\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{J}}_2\left(\omega \right)}\right)}\hfill \\ {\displaystyle -\left.\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\beta -1}d{\mathfrak{I}}_2\left(\omega \right)}\right)\right]{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\left[{\displaystyle \Gamma \left(\alpha \right)-{\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right]{\int }_1^{\infty }\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left[{\displaystyle \Gamma \left(\alpha \right)-{\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right]{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_1\left(\omega \right)}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}\left[{\displaystyle \Gamma \left(\alpha \right)-{\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{I}}_1\left(\omega \right)}\right]{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{J}}_2\left(\omega \right)}\hfill \\ {\displaystyle +\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)}\right){\int }_1^{\infty }\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\alpha -1}\mathrm{h}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_1\left(\omega \right)}\right)}\hfill \\ -\left.{\displaystyle \frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\omega )}^{\alpha -1}d{\mathfrak{J}}_1\left(\omega \right)}\right)\left({\displaystyle {\int }_1^{\infty }\left({\displaystyle {\int }_1^{\omega }{\left({\displaystyle ln\frac{\omega }{\xi }}\right)}^{\beta -1}\mathrm{k}\left(\xi \right)\frac{d\xi }{\xi }}\right)d{\mathfrak{I}}_2\left(\omega \right)}\right)}\right\}\hfill \\ {\displaystyle =B_0+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left[{\displaystyle -{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right.}\right.}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{s}}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\hfill \\ {\displaystyle +{\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\hfill \\ {\displaystyle -\left.{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right]\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left[{\displaystyle \frac{\Gamma \left(\alpha \right)}{\Gamma \left(\beta \right)}{\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right.}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle -\frac{\Gamma \left(\alpha \right)}{\Gamma \left(\beta \right)}{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\hfill \\ -\left.{\displaystyle \left.{\displaystyle \frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right]\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle =B_0+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left\{\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left(ln\frac{\xi }{\omega }\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right]\right.}\right.}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right){\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)}\hfill \\ {\displaystyle -\left.{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\left({\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)}\right\}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left\{{\displaystyle \frac{\Gamma \left(\alpha \right)}{\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right]}\right.}\hfill \\ {\displaystyle -\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right)\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right]}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\beta \right)}\left({\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right.}\hfill \\ -\left.{\displaystyle \left.\left.{\displaystyle {\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right]\right\}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle =B_0+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left\{{\displaystyle {\int }_1^{\infty }\left\{{\displaystyle \frac{a}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right]}\right.}\right.}\hfill \\ {\displaystyle +\left.{\displaystyle \frac{c}{\Gamma \left(\alpha \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)}\right]}\right\}\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left\{{\displaystyle \frac{a}{\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right]}\right.}\hfill \\ +\left.{\displaystyle \left.{\displaystyle \frac{c}{\Gamma \left(\beta \right)}\left[{\displaystyle {\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)-{\int }_{\omega }^{\infty }{\left({\displaystyle ln\frac{\xi }{\omega }}\right)}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)}\right]}\right\}\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right\}\hfill \\ {\displaystyle ={\int }_1^{\infty }{g}_{\beta }(\upsilon ,\omega )\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{(ln\upsilon )}^{\beta -1}}{\Delta }\left[{\displaystyle a{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\right.}\hfill \\ {\displaystyle +c{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)}\right)\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +a{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ +\left.{\displaystyle c{\int }_1^{\infty }\left({\displaystyle {\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)}\right)\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega }}\right]\hfill \\ {\displaystyle ={\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathrm{h}\left(\omega \right)\frac{d\omega }{\omega }+{\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathrm{k}\left(\omega \right)\frac{d\omega }{\omega },t\in [1,\infty ),}\hfill \end{array}$$

where ${\mathfrak{E}}_3$ and ${\mathfrak{E}}_4$ are given by (11). So we deduced the solution $(\Phi (\upsilon ),\Psi (\upsilon \left)\right),\upsilon \in [1,\infty )$ given by (10). □

The next lemma can be easily obtained by utilizing the definitions of the functions $g_{\alpha }$, $g_{\beta }$, and ${\mathfrak{E}}_i$, where $i=1, \dots , 4$.

Lemma 5.

Assume that the functions ${\mathfrak{I}}_1,{\mathfrak{I}}_2,{\mathfrak{J}}_1,{\mathfrak{J}}_2$ are nondecreasing functions, $a,d\in {\mathbb{R}}_{+}$, $b,c\in \mathbb{R}$, and $\Delta >0$, and let $\theta >1$. Then, the functions $g_{\alpha },g_{\beta },$ and ${\mathfrak{E}}_i$, $i=1, \dots , 4$ have the following properties:

(a) The functions $g_{\alpha }$ and $g_{\beta }$ are continuous on $[1,\infty )\times [1,\infty )$;

(b) ${\displaystyle 0\le g_{\alpha }(\upsilon ,\omega )\le \frac{1}{\Gamma \left(\alpha \right)}{(ln\upsilon )}^{\alpha -1},0\le g_{\beta }(\upsilon ,\omega )\le \frac{1}{\Gamma \left(\beta \right)}{(ln\upsilon )}^{\beta -1},\forall \upsilon ,\omega \in [1,\infty );}$

(c) ${\displaystyle 0\le \frac{g_{\alpha }(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\le \frac{1}{\Gamma \left(\alpha \right)},0\le \frac{g_{\beta }(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\le \frac{1}{\Gamma \left(\beta \right)},\forall \upsilon ,\omega \in [1,\infty );}$

(d) The functions ${\mathfrak{E}}_i,i=1, \dots , 4$ are continuous on $[1,\infty )\times [1,\infty )$;

(e) ${\mathfrak{E}}_i(\upsilon ,\omega )\ge 0$ for all $(\upsilon ,\omega )\in [1,\infty )\times [1,\infty )$ and $i=1, \dots , 4$;

(f) ${\displaystyle \frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\le \frac{1}{\Gamma \left(\alpha \right)}+\frac{1}{\Delta \Gamma \left(\alpha \right)}\left({\displaystyle d{\int }_1^{\infty }{(ln\zeta )}^{\alpha -1}d{\mathfrak{I}}_1\left(\zeta \right)+b{\int }_1^{\infty }{(ln\zeta )}^{\alpha -1}d{\mathfrak{J}}_1\left(\zeta \right)}\right)}$

• ${\displaystyle =\frac{d}{\Delta }=:{\Lambda }_1>0,\forall \upsilon ,\omega \in [1,\infty )}$;

(g) ${\displaystyle \frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\le \frac{1}{\Delta \Gamma \left(\beta \right)}\left({\displaystyle d{\int }_1^{\infty }{(ln\zeta )}^{\beta -1}d{\mathfrak{I}}_2\left(\zeta \right)+b{\int }_1^{\infty }{(ln\zeta )}^{\beta -1}d{\mathfrak{J}}_2\left(\zeta \right)}\right)}$

• ${\displaystyle =\frac{b}{\Delta }=:{\Lambda }_2\ge 0,\forall \upsilon ,\omega \in [1,\infty )}$;

(h) ${\displaystyle \frac{{\mathfrak{E}}_3(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\le \frac{1}{\Delta \Gamma \left(\alpha \right)}\left({\displaystyle c{\int }_1^{\infty }{(ln\zeta )}^{\alpha -1}d{\mathfrak{I}}_1\left(\zeta \right)+a{\int }_1^{\infty }{(ln\zeta )}^{\alpha -1}d{\mathfrak{J}}_1\left(\zeta \right)}\right)}$

• ${\displaystyle =\frac{c}{\Delta }=:{\Lambda }_3\ge 0,\forall \upsilon ,\omega \in [1,\infty )}$;

(i) ${\displaystyle \frac{{\mathfrak{E}}_4(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\le \frac{1}{\Gamma \left(\beta \right)}+\frac{1}{\Delta \Gamma \left(\beta \right)}\left({\displaystyle c{\int }_1^{\infty }{(ln\zeta )}^{\beta -1}d{\mathfrak{I}}_2\left(\zeta \right)+a{\int }_1^{\infty }{(ln\zeta )}^{\beta -1}d{\mathfrak{J}}_2\left(\zeta \right)}\right)}$

• ${\displaystyle =\frac{a}{\Delta }=:{\Lambda }_4>0,\forall \upsilon ,\omega \in [1,\infty )}$;

(j) ${\displaystyle \underset{\upsilon \in [\theta ,\infty )}{min}\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\ge \frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}}$

• $\times \left({\displaystyle d{\int }_1^{\infty }{g}_{\alpha }(\zeta ,s)d{\mathfrak{I}}_1\left(\zeta \right)+b{\int }_1^{\infty }{g}_{\alpha }(\zeta ,s)d{\mathfrak{J}}_1\left(\zeta \right)}\right),$ $\forall \omega \in [1,\infty )$;

(k) ${\displaystyle \underset{\upsilon \in [\theta ,\infty )}{min}\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}=\frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}}$

• $\times \left({\displaystyle d{\int }_1^{\infty }{g}_{\beta }(\zeta ,s)d{\mathfrak{I}}_2\left(\zeta \right)+b{\int }_1^{\infty }{g}_{\beta }(\zeta ,s)d{\mathfrak{J}}_2\left(\zeta \right)}\right),$ $\forall \omega \in [1,\infty )$;

(l) ${\displaystyle \underset{\upsilon \in [\theta ,\infty )}{min}\frac{{\mathfrak{E}}_3(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}=\frac{{(ln\theta )}^{\beta -1}}{\Delta (1+{(ln\theta )}^{\beta -1})}}$

• $\times \left({\displaystyle c{\int }_1^{\infty }{g}_{\alpha }(\zeta ,s)d{\mathfrak{I}}_1\left(\zeta \right)+a{\int }_1^{\infty }{g}_{\alpha }(\zeta ,s)d{\mathfrak{J}}_1\left(\zeta \right)}\right),$ $\forall \omega \in [1,\infty )$;

(m) ${\displaystyle \underset{\upsilon \in [\theta ,\infty )}{min}\frac{{\mathfrak{E}}_4(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\ge \frac{{(ln\theta )}^{\beta -1}}{\Delta (1+{(ln\theta )}^{\beta -1})}}$

• $\times \left({\displaystyle c{\int }_1^{\infty }{g}_{\beta }(\zeta ,s)d{\mathfrak{I}}_2\left(\zeta \right)+a{\int }_1^{\infty }{g}_{\beta }(\zeta ,s)d{\mathfrak{J}}_2\left(\zeta \right)}\right),$ $\forall \omega \in [1,\infty )$.

Remark 1.

Under the assumptions of Lemma 5, one finds that $a,d>0$ and $b,c\ge 0$, so ${\Lambda }_1,{\Lambda }_4>0$ and ${\Lambda }_2,{\Lambda }_3\ge 0$.

3. Main Results

We introduce the space

$$X_1=\left\{{\displaystyle \Phi \in C(I,\mathbb{R}),\underset{\upsilon \in I}{sup}\frac{|\Phi (\upsilon \left)\right|}{1+{(ln\upsilon )}^{\alpha -1}}<\infty }\right\},$$

where $I=[1,\infty )$, with the norm ${\parallel \Phi \parallel }_1={sup}_{\upsilon \in I}\frac{|\Phi (\upsilon \left)\right|}{1+{(ln\upsilon )}^{\alpha -1}}$, the space

$$X_2=\left\{{\displaystyle \Psi \in C(I,\mathbb{R}),\underset{\upsilon \in I}{sup}\frac{|\Psi (\upsilon \left)\right|}{1+{(ln\upsilon )}^{\beta -1}}<\infty }\right\},$$

with the norm ${\parallel \Psi \parallel }_2={sup}_{\upsilon \in I}\frac{|\Psi (\upsilon \left)\right|}{1+{(ln\upsilon )}^{\beta -1}}$, and the space $X=X_1\times X_2$ with the norm $\parallel (\Phi ,\Psi )\parallel ={\parallel \Phi \parallel }_1+{\parallel \Psi \parallel }_2$. The spaces $(X_1,\parallel ·{\parallel }_1)$, $(X_2,\parallel ·{\parallel }_2)$ and $(X,\parallel (·,·)\parallel )$ are Banach spaces (see [3], Lemma 2.7).

Lemma 6

([3], Lemma 2.8). Let $\Omega \subset X_1$ be a bounded set, which satisfies the following conditions:

(i) The functions ${\displaystyle \frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}},\Phi \in \Omega }$ are equicontinuous on any compact interval of I;

(ii) For any $ϵ>0$, there exists a constant $T=T\left(ϵ\right)>0$ such that

$$\left|{\displaystyle \frac{\Phi \left({\upsilon }_1\right)}{1+{(ln{\upsilon }_1)}^{\alpha -1}}-\frac{\Phi \left({\upsilon }_2\right)}{1+{(ln{\upsilon }_2)}^{\alpha -1}}}\right|<ϵ,\forall {\upsilon }_1,{\upsilon }_2\ge T,\forall \Phi \in \Omega .$$

Then, Ω is relatively compact in $X_1$.

Let us now define the positive cone $\mathcal{P}\subset X$ by

$$\mathcal{P}=\left\{\right(\Phi ,\Psi )\in X,\Phi (\upsilon )\ge 0,\Psi (\upsilon )\ge 0,\forall \upsilon \in [1,\infty \left)\right\},$$

and the operator $\mathcal{A}:\mathcal{P}\to X$ by $\mathcal{A}(\Phi ,\Psi )=({\mathcal{A}}_1(\Phi ,\Psi ),{\mathcal{A}}_2(\Phi ,\Psi )),(\Phi ,\Psi )\in \mathcal{P}$, where the operators ${\mathcal{A}}_1:\mathcal{P}\to X_1$ and ${\mathcal{A}}_2:\mathcal{P}\to X_2$ are defined by

$$\begin{array}{c}{\displaystyle {\mathcal{A}}_1(\Phi ,\Psi )\left(\upsilon \right)={\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega },\upsilon \in I,}\hfill \\ {\displaystyle {\mathcal{A}}_2(\Phi ,\Psi )\left(\upsilon \right)={\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega },\upsilon \in I,}\hfill \end{array}$$

for $(\Phi ,\Psi )\in \mathcal{P}$.

Subsequently, we outline the fundamental assumptions that will serve as the foundation for our main results.

(H1)

$\alpha \in (n-1,n]$, $\beta \in (m-1,m]$, $n,m\in \mathbb{N}$, $n,m\ge 2$, ${\mathfrak{I}}_1,{\mathfrak{I}}_2,{\mathfrak{J}}_1,{\mathfrak{J}}_2:[1,\infty )\to \mathbb{R}$ are nondecreasing functions, $a,d\in {\mathbb{R}}_{+}$, $b,c\in \mathbb{R}$, and $\Delta >0$ (given by (6)).

(H2)

The functions $\mathfrak{L}$ and $\mathfrak{M}$ belong to the set of continuous functions $C(\mathbb{R}+\times \mathbb{R}+,\mathbb{R}+)$, and they are non-zero on every subinterval of $(0,\infty )\times (0,\infty )$. Furthermore, both $\mathfrak{L}$ and $\mathfrak{M}$ are bounded on the entire domain $\mathbb{R}+\times {\mathbb{R}}_{+}$.

(H3)

The functions $\mathfrak{c},\mathfrak{d}:[1,\infty )\to {\mathbb{R}}_{+}$ are non-zero on any subinterval of $[1,\infty )$, and $0<{\int }_1^{\infty }\frac{\mathfrak{c}\left(\omega \right)}{\omega }d\omega <\infty$, $0<{\int }_1^{\infty }\frac{\mathfrak{d}\left(\omega \right)}{\omega }d\omega <\infty$.

Lemma 7.

If $\left(H1\right)$–$\left(H3\right)$ hold, then the operator $\mathcal{A}:\mathcal{P}\to \mathcal{P}$ is completely continuous.

Proof. 

Under the assumptions of this lemma, one obtains $\mathcal{A}(\Phi ,\Psi )\in \mathcal{P}$ for all $(\Phi ,\Psi )\in \mathcal{P}$, that is, $\mathcal{A}:\mathcal{P}\to \mathcal{P}$. We will prove this lemma in four steps.

(I) We show firstly that the operator $\mathcal{A}$ is uniformly bounded on $\mathcal{P}$. Let $\mathcal{S}$ be a bounded set of $\mathcal{P}$. Then, there exists $r>0$ such that $\parallel (\Phi ,\Psi )\parallel \le r$, and so ${\parallel \Phi \parallel }_1\le r$ and ${\parallel \Psi \parallel }_2\le r$ for all $(\Phi ,\Psi )\in \mathcal{S}$. By $\left(H2\right)$, there exist $M_1>0$ and $M_2>0$ such that $\mathfrak{L}(\Phi ,\Psi )\le M_1$ and $\mathfrak{M}(\Phi ,\Psi )\le M_2$ for all $\Phi ,\Psi \in {\mathbb{R}}_{+}$. Then, by $\left(H3\right)$, for any $(\Phi ,\Psi )\in \mathcal{S}$, one obtains

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1=\underset{\upsilon \in I}{sup}\frac{|{\mathcal{A}}_1(\Phi ,\Psi )\left(\upsilon \right)|}{1+{(ln\upsilon )}^{\alpha -1}}}\hfill \\ {\displaystyle =\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}\left({\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right.}\hfill \\ +\left.{\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right)\hfill \\ {\displaystyle \le M_1{\Lambda }_1{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+M_2{\Lambda }_2{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }=:M_3<\infty ,}\hfill \\ {\displaystyle \parallel {\mathcal{A}}_2{(\Phi ,\Psi )\parallel }_2=\underset{\upsilon \in I}{sup}\frac{|{\mathcal{A}}_2(\Phi ,\Psi )\left(\upsilon \right)|}{1+{(ln\upsilon )}^{\beta -1}}}\hfill \\ {\displaystyle =\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\beta -1}}\left({\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right.}\hfill \\ +\left.{\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right)\hfill \\ {\displaystyle \le M_1{\Lambda }_3{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+M_2{\Lambda }_4{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }=:M_4<\infty .}\hfill \end{array}$$

So,

$$\parallel \mathcal{A}(\Phi ,\Psi )\parallel =\parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1+{\parallel {\mathcal{A}}_2(\Phi ,\Psi )\parallel }_2\le M_3+M_4<\infty ,\forall (\Phi ,\Psi )\in \mathcal{S},$$

that is, operator $\mathcal{A}$ is uniformly bounded.

(II) We will now demonstrate the equicontinuity of $\mathcal{A}$ on every compact interval of I. Let us consider the interval $[1,T]$, where $T>1$. Then, for any ${\upsilon }_1,{\upsilon }_2\in [1,T]$, with ${\upsilon }_1<{\upsilon }_2$ and $(\Phi ,\Psi )\in \mathcal{S}$, we have

$$\begin{array}{c}{\Xi }_1=\left|{\displaystyle \frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_2\right)}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_1\right)}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\hfill \\ =\left|{\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right.\hfill \\ {\displaystyle +{\int }_1^{\infty }\frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ -{\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ \left.{\displaystyle -{\int }_1^{\infty }\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right|\hfill \\ \le \left|{\displaystyle {\int }_1^{\infty }\left({\displaystyle \frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right|\hfill \\ +\left|{\displaystyle {\int }_1^{\infty }\left({\displaystyle \frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right|\hfill \\ {\displaystyle \le {\int }_1^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{\infty }\left[\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\right.}\hfill \\ {\displaystyle +\frac{1}{\Delta }\left({\displaystyle d{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)+b{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right)}\hfill \\ {\displaystyle \left.\times \left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\right]\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\frac{1}{\Delta }\left({\displaystyle d{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)+b{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle \times \left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ \le M_1\underset{{\Xi }_0}{\underbrace{{\displaystyle {\int }_1^{\infty }\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}}}\hfill \\ {\displaystyle +\frac{M_1}{\Delta \Gamma \left(\alpha \right)}\left({\displaystyle d{\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)+b{\int }_1^{\infty }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)}\hfill \\ {\displaystyle \times \left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{M_2}{\Delta \Gamma \left(\beta \right)}\left({\displaystyle d{\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)+b{\int }_1^{\infty }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)}\hfill \\ {\displaystyle \times \left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =M_1{\Xi }_0+\frac{M_1(d\Gamma \left(\alpha \right)-\Delta )}{\Delta \Gamma \left(\alpha \right)}\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{M_{2}b}{\Delta }\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }.}\hfill \end{array}$$

For ${\Xi }_0$, one deduces

$$\begin{array}{c}{\displaystyle {\Xi }_0={\int }_1^{{\upsilon }_1}\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_{{\upsilon }_1}^{{\upsilon }_2}\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_{{\mathrm{t}}_2}^{\infty }\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{{\upsilon }_1}\left|{\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}-{\left(ln\frac{{\upsilon }_2}{\omega }\right)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}-{\left(ln\frac{{\upsilon }_1}{s}\right)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_{{\upsilon }_1}^{{\upsilon }_2}\left|{\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}-{\left(ln\frac{{\upsilon }_2}{\omega }\right)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_{{\upsilon }_2}^{\infty }\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le \frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_1^{{\upsilon }_1}\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_1^{{\upsilon }_1}\left({\displaystyle \frac{{\left(ln\frac{{\upsilon }_2}{\omega }\right)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\left(ln\frac{{\upsilon }_1}{\omega }\right)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right)\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_{{\upsilon }_1}^{{\upsilon }_2}\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}{\int }_{{\upsilon }_1}^{{\upsilon }_2}\frac{{\left(ln\frac{{\upsilon }_2}{s}\right)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{1}{\Gamma \left(\alpha \right)}\left({\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right){\int }_{{\upsilon }_2}^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }.}\hfill \end{array}$$

Because the functions $\frac{{(ln\upsilon )}^{\alpha -1}}{1+{(ln\upsilon )}^{\alpha -1}}$ and $\frac{{(ln(\upsilon /\omega ))}^{\alpha -1}}{1+{(ln\upsilon )}^{\alpha -1}}$ are uniformly continuous on $[1,T]$, respectively on $[1,T]\times [1,T]$, and the integrals ${\int }_0^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }$, ${\int }_0^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }$ are convergent, we conclude that ${\Xi }_0\to 0$ and ${\Xi }_1\to 0$ as ${\upsilon }_2\to {\upsilon }_1$ uniformly with respect to $(\Phi ,\Psi )\in \mathcal{S}$.

In a similar manner, one obtains

$$\left|{\displaystyle \frac{{\mathcal{A}}_2(\Phi ,\Psi )\left({\upsilon }_2\right)}{1+{(ln{\upsilon }_2)}^{\beta -1}}-\frac{{\mathcal{A}}_2(\Phi ,\Psi )\left({\upsilon }_1\right)}{1+{(ln{\upsilon }_1)}^{\beta -1}}}\right|\to 0,\mathrm{as}{\upsilon }_2\to {\upsilon }_1,$$

uniformly with respect to $(\Phi ,\Psi )\in \mathcal{S}$. Then, $\mathcal{A}\left(\mathcal{S}\right)$ is equicontinuous on $[1,T]$.

(III) In what follows, we will show that $\mathcal{A}$ is equiconvergent at ∞. We will prove firstly that ${\mathcal{A}}_1$ is equiconvergent at ∞, that is, for any $ϵ>0$ there exists $T_0>0$ such that for all ${\upsilon }_1,{\upsilon }_2\ge T_0$ and $(\Phi ,\Psi )\in \mathcal{S}$, one finds

$$\left|{\displaystyle \frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_2\right)}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_1\right)}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|<{\Lambda }_0ϵ,({\Lambda }_0>0).$$

For this, let $ϵ>0$. Then, there exists ${\delta }_1>1$ such that ${\int }_{{\delta }_1}^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }<ϵ$ and ${\int }_{{\delta }_1}^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }<ϵ$. Because ${lim}_{\upsilon \to \infty }\frac{{(ln\upsilon )}^{\alpha -1}}{1+{(ln\upsilon )}^{\alpha -1}}=1$ and ${lim}_{\upsilon \to \infty }\frac{g_{\alpha }(\upsilon ,{\delta }_1)}{1+{(ln\upsilon )}^{\alpha -1}}=0$, one deduces that there exist ${\delta }_2>0$ and ${\delta }_3>{\delta }_1$ such that for any ${\upsilon }_1,{\upsilon }_2>{\delta }_2$ we have

$$\left|{\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|<ϵ,$$

and for any ${\upsilon }_1,{\upsilon }_2>{\delta }_3$ and $1\le \omega \le {\delta }_1$, one finds

$$\begin{array}{c}\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\le \left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}}\right|+\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\hfill \\ \le \left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,{\delta }_1)}{1+{(ln{\upsilon }_2)}^{\alpha -1}}}\right|+\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_1,{\delta }_1)}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|<ϵ.\hfill \end{array}$$

Let us choose $T_0>max\{{\delta }_2,{\delta }_3\}$. Then, for any $(\Phi ,\Psi )\in \mathcal{S}$ and ${\upsilon }_1,{\upsilon }_2\ge T_0$, one obtains

$$\begin{array}{c}\left|{\displaystyle \frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_2\right)}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathcal{A}}_1(\Phi ,\Psi )\left({\upsilon }_1\right)}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\hfill \\ {\displaystyle \le {\int }_1^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{{\delta }_1}\left|{\displaystyle \frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_{{\delta }_1}^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_1({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_1({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\mathfrak{L}(x\left(\omega \right),y\left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{{\delta }_1}\left|{\displaystyle \frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_{{\delta }_1}^{\infty }\left|{\displaystyle \frac{{\mathfrak{E}}_2({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{\mathfrak{E}}_2({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le M_1{\int }_1^{{\delta }_1}\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +M_1{\int }_{{\delta }_1}^{\infty }\left|{\displaystyle \frac{g_{\alpha }({\upsilon }_2,\omega )}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{g_{\alpha }({\upsilon }_1,\omega )}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{M_1}{\Delta }\left|{\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|}\hfill \\ {\displaystyle \times \left({\displaystyle d{\int }_1^{\infty }\frac{1}{\Gamma \left(\alpha \right)}{\xi }^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)+b{\int }_1^{\infty }\frac{1}{\Gamma \left(\alpha \right)}{\xi }^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right){\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{M_2}{\Delta }\left|{\displaystyle \frac{{(ln{\upsilon }_2)}^{\alpha -1}}{1+{(ln{\upsilon }_2)}^{\alpha -1}}-\frac{{(ln{\upsilon }_1)}^{\alpha -1}}{1+{(ln{\upsilon }_1)}^{\alpha -1}}}\right|}\hfill \\ {\displaystyle \times \left({\displaystyle d{\int }_1^{\infty }\frac{1}{\Gamma \left(\beta \right)}{\xi }^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)+b{\int }_1^{\infty }\frac{1}{\Gamma \left(\beta \right)}{\xi }^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right){\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le ϵM_1{\int }_1^{{\delta }_1}\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+\frac{2ϵM_1}{\Gamma \left(\alpha \right)}+\frac{ϵM_1(d\Gamma \left(\alpha \right)-\Delta )}{\Delta \Gamma \left(\alpha \right)}{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{ϵM_{2}b}{\Delta }{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }={\Lambda }_0ϵ,({\Lambda }_0>0).}\hfill \end{array}$$

So, ${\mathcal{A}}_1$ is equiconvergent at ∞. In a similar manner, we show that ${\mathcal{A}}_2$ is equiconvergent at ∞, and then one deduces that $\mathcal{A}$ is equiconvergent at ∞.

(IV) In the final part of the proof, we will establish the continuity of the operator $\mathcal{A}$. Let ${\left(({\Phi }_n,{\Psi }_n)\right)}_{n\in \mathbb{N}},(\Phi ,\Psi )\in X$, $({\Phi }_n,{\Psi }_n)\to (\Phi ,\Psi )$ in $X=X_1\times X_2$, for $n\to \infty$, that is,

$${\displaystyle \underset{\upsilon \in I}{sup}\frac{|{\Phi }_n\left(\upsilon \right)-\Phi \left(\upsilon \right)|}{1+{(ln\upsilon )}^{\alpha -1}}\to 0,\mathrm{and}\underset{\upsilon \in I}{sup}\frac{|{\Psi }_n\left(\upsilon \right)-\Psi \left(\upsilon \right)|}{1+{(ln\upsilon )}^{\beta -1}}\to 0,\mathrm{as}n\to \infty .}$$

Then, one deduces that for any $\omega \in I$, ${\Phi }_n\left(\omega \right)-\Phi \left(\omega \right)\to 0$ and ${\Psi }_n\left(\omega \right)-\Psi \left(\omega \right)\to 0$, as $n\to \infty$.

Because

$$\begin{array}{c}|\mathfrak{L}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\le 2M_1,\forall \omega \in I,n\in \mathbb{N},\hfill \\ |\mathfrak{M}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\le 2M_2,\forall \omega \in I,n\in \mathbb{N},\hfill \end{array}$$

one obtains by the Lebesgue convergence theorem that

$$\begin{array}{c}{\displaystyle {\int }_1^{\infty }\mathfrak{c}\left(\omega \right)|\mathfrak{L}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }\to 0,\mathrm{as}n\to \infty ,}\hfill \\ {\displaystyle {\int }_1^{\infty }\mathfrak{d}\left(\omega \right)|\mathfrak{M}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }\to 0,\mathrm{as}n\to \infty .}\hfill \end{array}$$

(13)

By using Lemma 5 and (13), one finds

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1({\Phi }_n,{\Psi }_n)-{\mathcal{A}}_1(\Phi ,\Psi ){\parallel }_1=\underset{\upsilon \in I}{sup}\frac{|{\mathcal{A}}_1({\Phi }_n,{\Psi }_n)\left(\upsilon \right)-{\mathcal{A}}_1(\Phi ,\Psi )\left(\upsilon \right)|}{1+{(ln\upsilon )}^{\alpha -1}}}\hfill \\ {\displaystyle =\underset{\upsilon \in I}{sup}\left|{\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))\frac{d\omega }{\omega }}\right.}\hfill \\ {\displaystyle +{\int }_1^{\infty }\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ -{\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ \left.{\displaystyle -{\int }_1^{\infty }\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right|\hfill \\ {\displaystyle \le \underset{\upsilon \in I}{sup}\left({\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)|\mathfrak{L}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }}\right.}\hfill \\ +\left.{\displaystyle {\int }_1^{\infty }\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)|\mathfrak{M}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }}\right)\hfill \\ {\displaystyle \le {\Lambda }_1{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)|\mathfrak{L}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\Lambda }_2{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)|\mathfrak{M}({\Phi }_n\left(\omega \right),{\Psi }_n\left(\omega \right))-\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))|\frac{d\omega }{\omega }\to 0,\mathrm{as}n\to \infty .}\hfill \end{array}$$

Then, we deduce that $\parallel {\mathcal{A}}_1({\Phi }_n,{\Psi }_n)-{\mathcal{A}}_1(\Phi ,\Psi ){\parallel }_1\to 0,$ as $n\to \infty$. Using a similar approach, we can demonstrate that $\parallel {\mathcal{A}}_2({\Phi }_n,{\Psi }_n)-{\mathcal{A}}_2(\Phi ,\Psi ){\parallel }_2\to 0,$ as $n\to \infty$. Therefore, one obtains $\parallel \mathcal{A}({\Phi }_n,{\Psi }_n)-\mathcal{A}(\Phi ,\Psi )\parallel \to 0$ as $n\to \infty$. So, the operator $\mathcal{A}$ is continuous.

By combining Lemma 6 with the aforementioned steps, one deduces that the operator $\mathcal{A}$ is indeed completely continuous. □

For $\theta >1$, let us now introduce the following constants:

$$\begin{array}{c}{\displaystyle Q_1={\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega },Q_2={\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega },Q_3={\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega },Q_4={\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega },}\hfill \\ {\displaystyle {\tilde{L}}_1=\frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\zeta )}^{\alpha -1}d{\mathfrak{I}}_1\left(\zeta \right)+\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\zeta )}^{\alpha -1}d{\mathfrak{J}}_1\left(\zeta \right),}\hfill \\ {\displaystyle {\tilde{L}}_2=\frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\zeta )}^{\beta -1}d{\mathfrak{I}}_2\left(\zeta \right)+\frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\zeta )}^{\beta -1}d{\mathfrak{J}}_2\left(\zeta \right),}\hfill \\ {\displaystyle {\tilde{L}}_3=\frac{c}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\zeta )}^{\alpha -1}d{\mathfrak{I}}_1\left(\zeta \right)+\frac{a}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\zeta )}^{\alpha -1}d{\mathfrak{J}}_1\left(\zeta \right),}\hfill \\ {\displaystyle {\tilde{L}}_4=\frac{c}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\zeta )}^{\beta -1}d{\mathfrak{I}}_2\left(\zeta \right)+\frac{a}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\zeta )}^{\beta -1}d{\mathfrak{J}}_2\left(\zeta \right),}\hfill \\ {\displaystyle L_1=\frac{{\tilde{L}}_1{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})},L_2=\frac{{\tilde{L}}_2{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})},}\hfill \\ {\displaystyle L_3=\frac{{\tilde{L}}_3{(ln\theta )}^{\beta -1}}{\Delta (1+{(ln\theta )}^{\beta -1})},L_4=\frac{{\tilde{L}}_4{(ln\theta )}^{\beta -1}}{\Delta (1+{(ln\theta )}^{\beta -1})},}\hfill \\ {{\rm Y}}_1=Q_3(L_1+L_3),{{\rm Y}}_2=Q_4(L_2+L_4),{{\rm Y}}_3=Q_1({\Lambda }_1+{\Lambda }_3),{{\rm Y}}_4=Q_2({\Lambda }_2+{\Lambda }_4).\hfill \end{array}$$

(14)

Our first main theorem establishes the existence of at least one positive solution to the problem (1),(2) by employing the Guo–Krasnosel’skii fixed point theorem.

Theorem 1.

Assume that assumptions $\left(H1\right)$–$\left(H3\right)$ hold, and there exists $\theta >1$ such that $Q_j>0$, $j=3,4$, and ${\tilde{L}}_i>0$, $i=1,\dots ,4$. In addition, suppose that there exist positive constants $r_1,r_2$ with $r_1<r_2$, and ${\sigma }_1\in [{{\rm Y}}_1^{-1},\infty )$, ${\sigma }_2\in [{{\rm Y}}_2^{-1},\infty )$, ${\sigma }_3\in (0,{{\rm Y}}_3^{-1}]$ and ${\sigma }_4\in (0,{{\rm Y}}_4^{-1}]$ such that

(H4)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\ge \frac{{\sigma }_1{r}_1}{2},\forall \upsilon \in [\theta ,\infty ),(\varphi ,\psi )\in [0,r_1]\times [0,r_1],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\ge \frac{{\sigma }_2{r}_1}{2},\forall \upsilon \in [\theta ,\infty ),(\varphi ,\psi )\in [0,r_1]\times [0,r_1];$

(H5)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{{\sigma }_3{r}_2}{2},\forall \upsilon \in I,(\varphi ,\psi )\in [0,r_2]\times [0,r_2],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{{\sigma }_4{r}_2}{2},\forall \upsilon \in I,(\varphi ,\psi )\in [0,r_2]\times [0,r_2].$

Then, the problem (1),(2) has at least one positive solution $(\Phi ,\Psi )$ such that $r_1\le \parallel (\Phi ,\Psi )\parallel \le r_2.$

Proof. 

The operator $\mathcal{A}$ is proven to be completely continuous according to Lemma 7. We introduce the set ${\Omega }_1=\{(\Phi ,\Psi )\in X,\parallel (\Phi ,\Psi )\parallel <r_1\}$. Then, for $(\Phi ,\Psi )\in \mathcal{P}\cap \partial {\Omega }_1$, one obtains ${\parallel \Phi \parallel }_1+{\parallel \Psi \parallel }_2=r_1$, so ${\parallel \Phi \parallel }_1\le r_1$ and ${\parallel \Psi \parallel }_2\le r_1$, that is, $0\le \frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}\le r_1$ and $0\le \frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\le r_1$ for all $\upsilon \in I$.

Therefore, by $\left(H4\right)$, Lemma 5 and (14), one finds

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1=\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}\left({\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right.}\hfill \\ \left.{\displaystyle +{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right)\hfill \\ {\displaystyle \ge \underset{\upsilon \in [\theta ,\infty )}{inf}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}\left({\displaystyle {\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right.}\hfill \\ \left.{\displaystyle +{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\right)\hfill \\ {\displaystyle \ge \underset{\upsilon \in [\theta ,\infty )}{inf}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\underset{\upsilon \in [\theta ,\infty )}{inf}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \ge {\int }_1^{\infty }\underset{\upsilon \in [\theta ,\infty )}{inf}\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\underset{\upsilon \in [\theta ,\infty )}{inf}\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \ge \frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\left({\displaystyle d{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)}\right.}\hfill \\ {\displaystyle \left.{\displaystyle +b{\int }_1^{\infty }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right)\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\left({\displaystyle d{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)}\right.}\hfill \\ {\displaystyle \left.{\displaystyle +b{\int }_1^{\infty }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right)\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \ge \frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\left({\displaystyle d{\int }_1^{\theta }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{I}}_1\left(\xi \right)}\right.}\hfill \\ {\displaystyle \left.{\displaystyle +b{\int }_1^{\theta }{g}_{\alpha }(\xi ,\omega )d{\mathfrak{J}}_1\left(\xi \right)}\right)\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\left({\displaystyle d{\int }_1^{\theta }{g}_{\beta }(\xi ,\omega )d{\mathfrak{I}}_2\left(\xi \right)}\right.}\hfill \\ {\displaystyle \left.{\displaystyle +b{\int }_1^{\theta }{g}_{\beta }(\xi ,\omega )d{\mathfrak{J}}_2\left(\xi \right)}\right)\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ \times \left({\displaystyle \frac{d}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\xi )}^{\alpha -1}d{\mathfrak{I}}_1\left(\xi \right)+\frac{b}{\Gamma \left(\alpha \right)}{\int }_1^{\theta }{(ln\xi )}^{\alpha -1}d{\mathfrak{J}}_1\left(\xi \right)}\right)\hfill \\ {\displaystyle +\frac{{(ln\theta )}^{\alpha -1}}{\Delta (1+{(ln\theta )}^{\alpha -1})}{\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ \times \left({\displaystyle \frac{d}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\xi )}^{\beta -1}d{\mathfrak{I}}_2\left(\xi \right)+\frac{b}{\Gamma \left(\beta \right)}{\int }_1^{\theta }{(ln\xi )}^{\beta -1}d{\mathfrak{J}}_2\left(\xi \right)}\right)\hfill \\ {\displaystyle =\frac{{(ln\theta )}^{\alpha -1}{\tilde{L}}_1}{\Delta (1+{(ln\theta )}^{\alpha -1})}\frac{{\sigma }_1{r}_1}{2}{\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{(ln\theta )}^{\alpha -1}{\tilde{L}}_2}{\Delta (1+{(ln\theta )}^{\alpha -1})}\frac{{\sigma }_2{r}_1}{2}{\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\frac{{\sigma }_1{r}_1{L}_1{Q}_3}{2}+\frac{{\sigma }_2{r}_1{L}_2{Q}_4}{2}=\left({\displaystyle \frac{{\sigma }_1{L}_1{Q}_3}{2}+\frac{{\sigma }_2{L}_2{Q}_4}{2}}\right)r_1.}\hfill \end{array}$$

In a similar manner, one deduces

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_2{(\Phi ,\Psi )\parallel }_2\ge \frac{{(ln\theta )}^{\beta -1}{\tilde{L}}_3}{\Delta (1+{(ln\theta )}^{\beta -1})}\frac{{\sigma }_1{r}_1}{2}{\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{(ln\theta )}^{\beta -1}{\tilde{L}}_4}{\Delta (1+{(ln\theta )}^{\beta -1})}\frac{{\sigma }_2{r}_1}{2}{\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\frac{{\sigma }_1{r}_1{L}_3{Q}_3}{2}+\frac{{\sigma }_2{r}_1{L}_4{Q}_4}{2}=\left({\displaystyle \frac{{\sigma }_1{L}_3{Q}_3}{2}+\frac{{\sigma }_2{L}_4{Q}_4}{2}}\right)r_1.}\hfill \end{array}$$

Therefore, one finds

$$\begin{array}{c}\parallel \mathcal{A}(\Phi ,\Psi )\parallel =\parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1+{\parallel {\mathcal{A}}_2(\Phi ,\Psi )\parallel }_2\hfill \\ \ge \left[{\displaystyle \frac{(L_1+L_3){\sigma }_1{Q}_3}{2}+\frac{(L_2+L_4){\sigma }_2{Q}_4}{2}}\right]r_1=\left({\displaystyle \frac{{\sigma }_1{{\rm Y}}_1}{2}+\frac{{\sigma }_2{{\rm Y}}_2}{2}}\right)r_1\ge r_1,\hfill \end{array}$$

which gives us

$$\parallel \mathcal{A}(\Phi ,\Psi )\parallel \ge \parallel (\Phi ,\Psi )\parallel ,\forall (\Phi ,\Psi )\in \mathcal{P}\cap \partial {\Omega }_1.$$

(15)

Now, let us introduce the set ${\Omega }_2=\{(\Phi ,\Psi )\in X,\parallel (\Phi ,\Psi )\parallel <r_2\}$. For any $(\Phi ,\Psi )\in \mathcal{P}\cap \partial {\Omega }_2$, we have ${\parallel \Phi \parallel }_1+{\parallel \Psi \parallel }_2=r_2$, and then ${\parallel \Phi \parallel }_1\le r_2$ and ${\parallel \Psi \parallel }_2\le r_2$, that is, $0\le \frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}\le r_2$ and $0\le \frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\le r_2$ for all $\upsilon \in I$.

Then, by $\left(H5\right)$ and Lemma 5, one obtains

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1=\underset{\upsilon \in I}{sup}\frac{{\mathcal{A}}_1(\Phi ,\Psi )\left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}}\hfill \\ {\displaystyle \le \underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_1{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+{\Lambda }_2{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le \frac{{\sigma }_3{r}_2{\Lambda }_1}{2}{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{\sigma }_4{r}_2{\Lambda }_2}{2}{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ =\left({\displaystyle \frac{{\sigma }_3{Q}_1{\Lambda }_1}{2}+\frac{{\sigma }_4{Q}_2{\Lambda }_2}{2}}\right)r_2,\hfill \\ {\displaystyle \parallel {\mathcal{A}}_2{(\Phi ,\Psi )\parallel }_2=\underset{\upsilon \in I}{sup}\frac{{\mathcal{A}}_2(\Phi ,\Psi )\left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}}\hfill \\ {\displaystyle \le \underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\beta -1}}{\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\beta -1}}{\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_3(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_4(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_3{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+{\Lambda }_4{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le \frac{{\sigma }_3{r}_2{\Lambda }_3}{2}{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+\frac{{\sigma }_4{r}_2{\Lambda }_4}{2}{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ =\left({\displaystyle \frac{{\sigma }_3{Q}_1{\Lambda }_3}{2}+\frac{{\sigma }_4{Q}_2{\Lambda }_4}{2}}\right)r_2.\hfill \end{array}$$

Then, one deduces

$$\begin{array}{c}\parallel \mathcal{A}(\Phi ,\Psi )\parallel \le \left[{\displaystyle \frac{({\Lambda }_1+{\Lambda }_3){\sigma }_3{Q}_1}{2}+\frac{({\Lambda }_2+{\Lambda }_4){\sigma }_4{Q}_2}{2}}\right]r_2\hfill \\ =\left({\displaystyle \frac{{\sigma }_3{{\rm Y}}_3}{2}+\frac{{\sigma }_4{{\rm Y}}_4}{2}}\right)r_2\le r_2,\hfill \end{array}$$

that is,

$$\parallel \mathcal{A}(\Phi ,\Psi )\parallel \le \parallel (\Phi ,\Psi )\parallel ,\forall (\Phi ,\Psi )\in \mathcal{P}\cap \partial {\Omega }_2.$$

(16)

Hence, based on relations (15) and (16), as well as the Guo–Krasnosels’kii fixed point theorem (see [26]), it can be deduced that the operator $\mathcal{A}$ possesses a fixed point $(\Phi ,\Psi )\in \mathcal{P}\cap ({\overline{\Omega }}_2\setminus {\Omega }_1)$, where $r_1\le \left|(\Phi ,\Psi )\right|\le r_2$. So, $\Phi \left(\upsilon \right)\ge 0$ and $\Psi \left(\upsilon \right)\ge 0$ for all $\upsilon \in I$, and by $\left(H2\right)$,$\left(H3\right)$, one obtains that $\Phi \left(\upsilon \right)>0$ for all $\upsilon \in (1,\infty )$ or $\Psi \left(\upsilon \right)>0$ for all $\upsilon \in (1,\infty )$. Consequently, the pair $(\Phi ,\Psi )$ serves as a positive solution to the problem (1),(2). □

By employing analogous reasoning as in the proof of Theorem 1, we can derive the following theorem.

Theorem 2.

Assume that assumptions $\left(H1\right)$–$\left(H3\right)$ hold, and there exists $\theta >1$ such that $Q_j>0$, $j=3,4$, and ${\tilde{L}}_i>0$, $i=1, \dots , 4$. In addition, suppose that there exist positive constants $r_1,r_2$ with $r_1<r_2$, and ${\sigma }_1\in [{{\rm Y}}_1^{-1},\infty )$, ${\sigma }_2\in [{{\rm Y}}_2^{-1},\infty )$, ${\sigma }_3\in (0,{{\rm Y}}_3^{-1}]$ and ${\sigma }_4\in (0,{{\rm Y}}_4^{-1}]$ such that

(H6)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{{\sigma }_3{r}_1}{2},\forall \upsilon \in I,(\varphi ,\psi )\in [0,r_1]\times [0,r_1],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{{\sigma }_4{r}_1}{2},\forall \upsilon \in I,(\varphi ,\psi )\in [0,r_1]\times [0,r_1];$

(H7)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\ge \frac{{\sigma }_1{r}_2}{2},\forall \upsilon \in [\theta ,\infty ),(\varphi ,\psi )\in [0,r_2]\times [0,r_2],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\ge \frac{{\sigma }_2{r}_2}{2},\forall \upsilon \in [\theta ,\infty ),(\varphi ,\psi )\in [0,r_2]\times [0,r_2].$

Then, the problem (1),(2) has at least one positive solution $(\Phi ,\Psi )$ such that $r_1\le \parallel (\Phi ,\Psi )\parallel \le r_2.$

In what follows, we will prove the existence of at least three positive solutions for the problem (1),(2) by applying the Leggett–Williams fixed point theorem (Theorem 3.3 from [27]).

Theorem 3.

Assume that assumptions $\left(H1\right)$–$\left(H3\right)$ hold, and there exists $\theta >1$ such that $Q_j>0$, $j=3,4$, and ${\tilde{L}}_i>0$, $i=1, \dots , 4$. In addition, suppose that there exist positive constants $a_0<b_0<c_0$ such that

(H8)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )<\frac{a_0}{2{{\rm Y}}_3},\forall \upsilon \in I,(\varphi ,\psi )\in [0,a_0]\times [0,a_0],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )<\frac{a_0}{2{{\rm Y}}_4},\forall \upsilon \in I,(\varphi ,\psi )\in [0,a_0]\times [0,a_0];$

(H9)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )>\frac{b_0}{2{{\rm Y}}_1},\forall \upsilon \in [\theta ,\infty ),\varphi ,\psi \ge 0,b_0\le \varphi +\psi \le c_0,$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )>\frac{b_0}{2{{\rm Y}}_2},\forall \upsilon \in [\theta ,\infty ),\varphi ,\psi \ge 0,b_0\le \varphi +\psi \le c_0;$

(H10)

$\mathfrak{L}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{c_0}{2{{\rm Y}}_3},\forall \upsilon \in I,(\varphi ,\psi )\in [0,c_0]\times [0,c_0],$

$\mathfrak{M}((1+{(ln\upsilon )}^{\alpha -1})\varphi ,(1+{(ln\upsilon )}^{\beta -1})\psi )\le \frac{c_0}{2{{\rm Y}}_4},\forall \upsilon \in I,(\varphi ,\psi )\in [0,c_0]\times [0,c_0].$

Then, the problem (1),(2) has at least three positive solutions, $({\Phi }_1,{\Psi }_1)$, $({\Phi }_2,{\Psi }_2)$ and $({\Phi }_3,{\Psi }_3)$, such that $\parallel ({\Phi }_1,{\Psi }_1)\parallel <a_0$, $\parallel ({\Phi }_3,{\Psi }_3)\parallel >a_0$, and

$$\underset{\upsilon \in [\theta ,\infty )}{inf}\left(\frac{{\Phi }_2\left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{{\Psi }_2\left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\right)>b_0,\underset{\upsilon \in [\theta ,\infty )}{inf}\left(\frac{{\Phi }_3\left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{{\Psi }_3\left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\right)<b_0.$$

Proof. 

We show firstly that operator $\mathcal{A}:{\overline{\mathcal{P}}}_{c_0}\to {\overline{\mathcal{P}}}_{c_0}$. For any $(\Phi ,\Psi )\in {\overline{\mathcal{P}}}_{c_0}$, we have $\parallel (\Phi ,\Psi )\parallel \le c_0$, and so ${\parallel \Phi \parallel }_1\le c_0$, ${\parallel \Psi \parallel }_2\le c_0$. Using the assumption $\left(H10\right)$ and Lemma 5, one finds

$$\begin{array}{c}{\displaystyle \parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1\le \underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_1(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\alpha -1}}{\int }_1^{\infty }{\mathfrak{E}}_2(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_1(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_2(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\alpha -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_1{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+{\Lambda }_2{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_1\frac{c_0}{2{{\rm Y}}_3}{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+{\Lambda }_2\frac{c_0}{2{{\rm Y}}_4}{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\left({\displaystyle \frac{{\Lambda }_1{Q}_1}{{{\rm Y}}_3}+\frac{{\Lambda }_2{Q}_2}{{{\rm Y}}_4}}\right)\frac{c_0}{2},}\hfill \\ {\displaystyle \parallel {\mathcal{A}}_2{(\Phi ,\Psi )\parallel }_2\le \underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\beta -1}}{\int }_1^{\infty }{\mathfrak{E}}_3(\upsilon ,\omega )\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +\underset{\upsilon \in I}{sup}\frac{1}{1+{(ln\upsilon )}^{\beta -1}}{\int }_1^{\infty }{\mathfrak{E}}_4(\upsilon ,\omega )\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_3(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +{\int }_1^{\infty }\underset{\upsilon \in I}{sup}\frac{{\mathfrak{E}}_4(\upsilon ,\omega )}{1+{(ln\upsilon )}^{\beta -1}}\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_3{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+{\Lambda }_4{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle \le {\Lambda }_3\frac{c_0}{2{{\rm Y}}_3}{\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }+{\Lambda }_4\frac{c_0}{2{{\rm Y}}_4}{\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }}\hfill \\ {\displaystyle =\left({\displaystyle \frac{{\Lambda }_3{Q}_1}{{{\rm Y}}_3}+\frac{{\Lambda }_4{Q}_2}{{{\rm Y}}_4}}\right)\frac{c_0}{2}.}\hfill \end{array}$$

Then, one obtains

$${\displaystyle \parallel \mathcal{A}(\Phi ,\Psi )\parallel \le \left[{\displaystyle \frac{({\Lambda }_1+{\Lambda }_3)Q_1}{{{\rm Y}}_3}+\frac{({\Lambda }_2+{\Lambda }_4)Q_2}{{{\rm Y}}_4}}\right]\frac{c_0}{2}=c_0,}$$

so $\mathcal{A}:{\overline{\mathcal{P}}}_{c_0}\to {\overline{\mathcal{P}}}_{c_0}$.

Let $d_0\in (b_0,c_0)$ be chosen, and let us define the concave non-negative continuous functional w on $\mathcal{P}$ as follows:

$${\displaystyle w(\Phi ,\Psi )=\underset{\upsilon \in [\theta ,\infty )}{inf}\left({\displaystyle \frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}}\right),(\Phi ,\Psi )\in \mathcal{P}.}$$

One can easily see that $w(\Phi ,\Psi )\le \parallel (\Phi ,\Psi )\parallel$ for all $(\Phi ,\Psi )\in {\overline{\mathcal{P}}}_{c_0}$.

Next, we will verify the conditions (i)–(iii) of Theorem 3.3 from [27]. We verify condition (ii) first. For $(\Phi ,\Psi )\in {\overline{\mathcal{P}}}_{a_0}$, we will show that $\parallel \mathcal{A}(\Phi ,\Psi )\parallel <a_0$. For this, let $(\Phi ,\Psi )\in {\overline{\mathcal{P}}}_{a_0}$. As in the above inequalities, one obtains

$${\displaystyle \parallel {\mathcal{A}}_1{(\Phi ,\Psi )\parallel }_1<\left({\displaystyle \frac{{\Lambda }_1{Q}_1}{{{\rm Y}}_3}+\frac{{\Lambda }_2{Q}_2}{{{\rm Y}}_4}}\right)\frac{a_0}{2},{\parallel {\mathcal{A}}_2(\Phi ,\Psi )\parallel }_2<\left({\displaystyle \frac{{\Lambda }_3{Q}_1}{{{\rm Y}}_3}+\frac{{\Lambda }_4{Q}_2}{{{\rm Y}}_4}}\right)\frac{a_0}{2},}$$

and so $\parallel \mathcal{A}(\Phi ,\Psi )\parallel <a_0$. Then, we have assumption (ii).

We verify condition (i) from Theorem 3.3 from [27]. Let us choose the element

$$({\Phi }_0\left(\upsilon \right),{\Psi }_0\left(\upsilon \right))=\left({\displaystyle \frac{b_0+d_0}{4}\left(1+{(ln\upsilon )}^{\alpha -1}\right),\frac{b_0+d_0}{4}(1+{(ln\upsilon )}^{\beta -1})}\right),\upsilon \in I.$$

Because ${\Phi }_0\left(\upsilon \right)\ge 0$, ${\Psi }_0\left(\upsilon \right)\ge 0$ for all $\upsilon \in I$, $\parallel ({\Phi }_0,{\Psi }_0)\parallel =\frac{b_0+d_0}{2}<d_0$ and $w({\Phi }_0,{\Psi }_0)=\frac{b_0+d_0}{2}>b_0$, one deduces that $({\Phi }_0,{\Psi }_0)\in \left\{(\Phi ,\Psi )\right|(\Phi ,\Psi )\in \mathcal{P}(w,b_0,d_0),w(\Phi ,\Psi )>b_0\}$. Now, let $(\Phi ,\Psi )\in \mathcal{P}(w,b_0,d_0)$, that is, $(\Phi ,\Psi )\in \mathcal{P}$, $w(\Phi ,\Psi )\ge b_0$ and $\parallel (\Phi ,\Psi )\parallel \le d_0$. So, one finds $\frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\le d_0$ for all $\upsilon \in I$, and ${inf}_{\upsilon \in [\theta ,\infty )}\left(\frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}\right)$$\ge b_0$. Then, by $\left(H9\right)$ and Lemma 5, and similar computations to those from the first part of the proof of Theorem 1, one deduces

$$\begin{array}{c}{\displaystyle w\left(\mathcal{A}(\Phi ,\Psi )\right)=\underset{\upsilon \in [\theta ,\infty )}{inf}\left({\displaystyle \frac{{\mathcal{A}}_1(\Phi ,\Psi )\left(\upsilon \right)}{1+{(ln\upsilon )}^{\alpha -1}}+\frac{{\mathcal{A}}_2(\Phi ,\Psi )\left(\upsilon \right)}{1+{(ln\upsilon )}^{\beta -1}}}\right)}\hfill \\ {\displaystyle \ge L_1{\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+L_2{\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle +L_3{\int }_{\theta }^{\infty }\mathfrak{c}\left(\omega \right)\mathfrak{L}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }+L_4{\int }_{\theta }^{\infty }\mathfrak{d}\left(\omega \right)\mathfrak{M}(\Phi \left(\omega \right),\Psi \left(\omega \right))\frac{d\omega }{\omega }}\hfill \\ {\displaystyle >L_1\frac{b_0{Q}_3}{2{{\rm Y}}_1}+L_2\frac{b_0{Q}_4}{2{{\rm Y}}_2}+L_3\frac{b_0{Q}_3}{2{{\rm Y}}_1}+L_4\frac{b_0{Q}_4}{2{{\rm Y}}_2}}\hfill \\ {\displaystyle =(L_1+L_3)\frac{b_0{Q}_3}{2{{\rm Y}}_1}+(L_2+L_4)\frac{b_0{Q}_4}{2{{\rm Y}}_2}=b_0.}\hfill \end{array}$$

Therefore, $w\left(\mathcal{A}(\Phi ,\Psi )\right)>b_0$, and we have assumption (i).

Now, we verify condition (iii) of Theorem 3.3 from [27], namely $w\left(\mathcal{A}(\Phi ,\Psi )\right)>b_0$ for $(\Phi ,\Psi )\in \mathcal{P}(w,b_0,c_0)$ and $\parallel \mathcal{A}(\Phi ,\Psi )\parallel >d_0$. So, let $(\Phi ,\Psi )\in \mathcal{P}(w,b_0,c_0)$ and $\parallel \mathcal{A}(\Phi ,\Psi )\parallel >d_0$. By arguments similar to those used earlier, one deduces that $w\left(\mathcal{A}(\Phi ,\Psi )\right)>b_0$, that is, assumption (iii) is satisfied.

Applying Theorem 3.3 from [27] along with assumptions $\left(H2\right)$ and $\left(H3\right)$, we can infer that the problem (1),(2) possesses at least three positive solutions: $({\Phi }_1,{\Psi }_1)$, $({\Phi }_2,{\Psi }_2)$, and $({\Phi }_3,{\Psi }_3)$. Moreover, these solutions satisfy the following properties: $|({\Phi }_1,{\Psi }_1)|<a_0$, $|({\Phi }_3,{\Psi }_3)|>a_0$, $w({\Phi }_2,{\Psi }_2)>b_0$, and $w({\Phi }_3,{\Psi }_3)<b_0$. □

4. An Example

Let $\alpha =\frac{5}{2}$, $\beta =\frac{10}{3}$, $n=3$, $m=4$, $\mathfrak{c}\left(\upsilon \right)=\frac{1}{{(\upsilon -1/2)}^{1/3}},$$\upsilon \in [1,\infty )$, $\mathfrak{d}\left(\upsilon \right)=\frac{\upsilon +2}{{(\upsilon -1/3)}^{5/4}}$, $\upsilon \in [1,\infty )$, $\mathfrak{L}(\varphi ,\psi )=6+e^{-|\varphi -\psi |}{sin}^2\left(\varphi \psi \right)$, $\varphi ,\psi \ge 0$, $\mathfrak{M}(\varphi ,\psi )=\frac{2}{3}+95e^{-\varphi \psi }{cos}^4(\varphi +\psi ),$$\varphi ,\psi \ge 0$, ${\mathfrak{I}}_1\left(\omega \right)=\{\frac{19}{110},\omega \in [1,3);\frac{\omega }{11}-\frac{1}{10},\omega \in [3,7);\frac{59}{110},\omega \in [7,10);\frac{116}{165},\omega \in [10,\infty )\}$, ${\mathfrak{I}}_2\left(\omega \right)=\{\frac{5}{136}{(\omega -1)}^{17/5},\omega \in [1,2);\frac{5}{136},\omega \in (2,\infty )\}$, ${\mathfrak{J}}_1\left(\omega \right)=\{1,\omega \in [1,\frac{5}{3});\frac{14}{13},\omega \in [\frac{5}{3},2);\frac{7}{290}{(\omega -2)}^{29/7}+\frac{14}{13},\omega \in [2,\frac{15}{6});\frac{7}{290}{\left(\frac{1}{2}\right)}^{29/7}+\frac{14}{13},\omega \in [\frac{15}{6},\infty )\}$, ${\mathfrak{J}}_2\left(\omega \right)=\{3,\omega \in [1,4);\frac{286}{95},\omega \in [4,9);\frac{{\omega }^2}{146}+\frac{34061}{13870},\omega \in [9,11);\frac{45556}{13870},\omega \in [11,\infty )\}$.

Let us consider the system of fractional differential equations represented by

$$\left\{\begin{array}{c}{\displaystyle {}^H{D}_{1+}^{5/2}\Phi \left(\upsilon \right)+\frac{1}{{(\upsilon -1/2)}^{1/3}}\left(6+e^{-|\Phi (\upsilon )-\Psi (\upsilon \left)\right|}{sin}^2(\Phi \left(\upsilon \right)\Psi \left(\upsilon \right))\right)=0,\upsilon \in (1,\infty ),}\hfill \\ {\displaystyle {}^H{D}_{1+}^{10/3}\Psi \left(\upsilon \right)+\frac{\upsilon +2}{{(\upsilon -1/3)}^{5/4}}\left(\frac{2}{3}+95e^{-\Phi \left(\upsilon \right)\Psi \left(\upsilon \right)}{cos}^4(\Phi \left(\upsilon \right)+\Psi \left(\upsilon \right))\right)=0,\upsilon \in (1,\infty ),}\hfill \end{array}\right.$$

(17)

subject to the boundary conditions

$$\left\{\begin{array}{c}\Phi \left(1\right)=\Phi \prime \left(1\right)=0,\Psi \left(1\right)=\Psi \prime \left(1\right)=\Psi ″\left(1\right)=0,\hfill \\ {\displaystyle {}^H{D}_{1+}^{3/2}\Phi (\infty )=\frac{1}{11}{\int }_3^7\Phi \left(\omega \right)d\omega +\frac{1}{6}\Phi \left(10\right)+\frac{1}{8}{\int }_1^2{(\omega -1)}^{12/5}\Psi \left(\omega \right)d\omega ,}\hfill \\ {\displaystyle {}^H{D}_{1+}^{7/3}\Psi (\infty )=\frac{1}{13}\Phi \left({\displaystyle \frac{5}{3}}\right)+\frac{1}{10}{\int }_2^{15/6}{(\omega -2)}^{22/7}\Phi \left(\omega \right)d\omega +\frac{1}{95}\Psi \left(4\right)+\frac{1}{73}{\int }_9^{11}\omega \Psi \left(\omega \right)d\omega .}\hfill \end{array}\right.$$

(18)

By using the Mathematica program, one obtains $a\approx 0.01753901$, $b\approx 0.01031779$, $c\approx 0.02920547$, $d\approx 0.83235873$, $\Delta \approx 0.01429742>0$, ${\int }_1^{\infty }\mathfrak{c}\left(\omega \right)\frac{d\omega }{\omega }\approx 3.15855472$, ${\int }_1^{\infty }\mathfrak{d}\left(\omega \right)\frac{d\omega }{\omega }\approx 6.53061854$. So, assumptions $\left(H1\right)-\left(H3\right)$ are satisfied. In addition, we take $\theta =2$, and we deduce ${\Lambda }_1=\frac{d}{\Delta }\approx 58.21742336$, ${\Lambda }_2=\frac{b}{\Delta }\approx 0.72165469$, ${\Lambda }_3=\frac{c}{\Delta }\approx 2.04270988$, ${\Lambda }_4=\frac{a}{\Delta }\approx 1.22672621$, ${\tilde{L}}_1\approx 0.00021797$, ${\tilde{L}}_2\approx 0.00309129$, ${\tilde{L}}_3\approx 0.00037053$, ${\tilde{L}}_4\approx 0.00010846$, $L_1\approx 0.00557882$, $L_2\approx 0.07911642$, $L_3\approx 0.00773204$, $L_4\approx 0.00226336$, $Q_1\approx 3.15855472$, $Q_2\approx 6.53061854$, $Q_3\approx 2.43620073$, $Q_4\approx 4.28276191$, ${{\rm Y}}_1\approx 0.03242794$, ${{\rm Y}}_2\approx 0.34853023$, ${{\rm Y}}_3\approx 190.33492843$, ${{\rm Y}}_4\approx 12.72413242$.

Let us consider $r_1=\frac{1}{3}$, $r_2=2800$, ${\sigma }_1=31>{{\rm Y}}_1^{-1}$, ${\sigma }_2=3>{{\rm Y}}_2^{-1}$, ${\sigma }_3=0.005<{{\rm Y}}_3^{-1}$ and ${\sigma }_4=0.07<{{\rm Y}}_4^{-1}$. Then, one obtains the inequalities

$$\begin{array}{c}\mathfrak{L}((1+{(ln\upsilon )}^{3/2})\varphi ,(1+{(ln\upsilon )}^{7/3})\psi )\ge 6>\frac{{\sigma }_1{r}_1}{2}=\frac{31}{6},\forall \upsilon \in [2,\infty ),\varphi ,\psi \in [0,\frac{1}{3}],\hfill \\ \mathfrak{M}((1+{(ln\upsilon )}^{3/2})\varphi ,(1+{(ln\upsilon )}^{7/3})\psi )\ge \frac{2}{3}>\frac{{\sigma }_2{r}_1}{2}=\frac{1}{2},\forall \upsilon \in [2,\infty ),\varphi ,\psi \in [0,\frac{1}{3}],\hfill \end{array}$$

so assumption $\left(H4\right)$ is satisfied. In addition, one finds

$$\begin{array}{c}\mathfrak{L}((1+{(ln\upsilon )}^{3/2})\varphi ,(1+{(ln\upsilon )}^{7/3})\psi )\le 7=\frac{{\sigma }_3{r}_2}{2},\forall \upsilon \in [1,\infty ),\varphi ,\psi \in [0,2800],\hfill \\ \mathfrak{M}((1+{(ln\upsilon )}^{3/2})\varphi ,(1+{(ln\upsilon )}^{7/3})\psi )\le \frac{287}{3}<\frac{{\sigma }_4{r}_2}{2}=98,\forall \upsilon \in [1,\infty ),\varphi ,\psi \in [0,2800],\hfill \end{array}$$

hence, assumption $\left(H5\right)$ is also satisfied. Therefore, by Theorem 1 we conclude that the problem (17),(18) has at least one positive solution $(\Phi (\upsilon ),\Psi (\upsilon \left)\right),\upsilon \in [1,\infty )$, with $\frac{1}{3}\le {sup}_{\upsilon \in [1,\infty )}\frac{\Phi \left(\upsilon \right)}{1+{(ln\upsilon )}^{3/2}}+{sup}_{\upsilon \in [1,\infty )}\frac{\Psi \left(\upsilon \right)}{1+{(ln\upsilon )}^{7/3}}\le 2800$.

5. Conclusions

In this paper, we investigated the system of nonlinear fractional differential equations (1) with Hadamard derivatives of various orders $\alpha \in (n-1,n]$ and $\beta \in (m-1,m]$, respectively, and non-negative nonlinearities, on the infinite interval $(1,\infty )$. The system (1) is supplemented with general nonlocal boundary conditions (2), where the unknown functions $\Phi$ and $\Psi$ in the point 1 and their derivatives until orders $n-2$ and $m-2$, respectively, are all 0, and the Hadamard derivatives of $\Phi$ and $\Psi$ of order $n-1$ and $m-1$ at ∞ are dependent on both Riemann–Liouville integrals of $\Phi$ and $\Psi$. Our problem generalizes the problem studied in [1], by considering here different orders for the fractional derivatives in the equations of system (1), and also a general form of the boundary conditions from (2) at ∞. Under some assumptions on the data of this problem, we gave firstly the solution of the associated linear boundary value problem, and the corresponding Green functions with their properties. Then, in the main section of the paper, we proved the existence of positive solutions of (1),(2) by applying the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem. An example which illustrates the main theorems is finally presented. Our results can be generalized in the future for other fractional derivatives in the system (1), and for some complex networks, such as those from paper [28].