A Study of a Nonlocal Coupled Integral Boundary Value Problem for Nonlinear Hilfer–Hadamard-Type Fractional Langevin Equations

Abstract

We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results obtained.

1. Introduction

Fractional differential equations find extensive applications in the mathematical modeling of several phenomena occurring in natural, technical and social sciences. Examples include immune systems [1], chaotic synchronization [2], diffusion processes [3,4], ecological systems [5], neural networks [6], financial economics, etc. In contrast to the classical derivative, different types of fractional derivatives exist, like Riemann–Liouville, Liouville–Caputo, Hadamard, Hilfer, etc. The Hilfer fractional derivative introduced in [7] is reduced into the Riemann–Liouville and Caputo fractional derivatives, respectively, for the smallest and largest values of its parameter. For further details on the Hilfer derivative, we refer the reader to [8,9], while some results on boundary value problems involving the Hilfer fractional derivate can be found in the review article [10]. The Hadamard fractional derivative [11] contains a logarithmic function in its definition, and Caputo–Hadamard and Hilfer–Hadamard are regarded as variants of this derivative. One notices that the Hilfer–Hadamard fractional derivative is specialized into the Hadamard and Caputo–Hadamard fractional derivatives for the smallest and largest values of its parameter, respectively. For a detailed account of the work on Hadamard fractional differential equations, inclusions and inequalities, we refer the reader to the text [12]. One can find interesting results on Caputo–Hadamard fractional boundary value problems in [13,14]. The authors studied Hilfer–Hadamard fractional differential equations with multi-point and integral boundary conditions in [15,16].

Langevin [17] presented Newton’s second law of motion for stochastic physics during his study of Brownian motion, which is known as the Langevin equation. He was of the view that his approach to Brownian motion was “infinitely more simple” than that offered by Einstein. Later, it was discovered that the Langevin equation failed to represent complex systems. This was the pretext for offering certain generalizations of the Langevin equation to describe physical phenomena in disordered regions [18], statistical physics [19], fluctuation–dissipation configurations [20], etc. A fractional counterpart of the Langevin equation was proposed in [21] by replacing the ordinary derivative it contains with a fractional counterpart, while a Langevin equation of two different fractional orders was presented in [22]. One can find other variants of the Langevin equation in [23,24]. For results on boundary value problems involving a fractional-order Langevin equation, we refer the reader to [25,26,27]. Considerable interest has also been shown in studying nonlinear systems of fractional-order Langevin equations involving different fractional derivatives and boundary conditions; for instance, see [28,29] and the references cited therein.

The notion of Ulam–Hyers stability [30,31] appears in several areas of research, such as functional equations [32], the Black–Scholes equation [33], etc. The Ulam’s stability of an integral boundary value problem of coupled fractional differential equations was studied in [34]. The authors of [35] discussed the Hyers–Ulam stability for fractional differential equations using the Mittag–Leffler kernel. Some results on the Ulam–Hyers stability of Langevin fractional differential equations can be found in [36,37].

In this paper, influenced by the work described in the preceding paragraphs, we introduce a new set of nonlocal integral boundary conditions and study a coupled system of Hilfer–Hadamard fractional Langevin equations supplemented with these conditions. Precisely, we develop the existence theory and Ulam–Hyers stability for the system

$$\left\{\begin{array}{c}{\displaystyle {}^{HH}D_{1^{+}}^{{\kappa }_1,{\xi }_1}({}^{HH}D_{1^{+}}^{{\kappa }_2,{\xi }_2}+{\chi }_1)x\left(t\right)=f_1(t,x\left(t\right),y\left(t\right)),t\in \mathcal{J},}\hfill \\ {\displaystyle {}^{HH}D_{1^{+}}^{{\kappa }_3,{\xi }_3}({}^{HH}D_{1^{+}}^{{\kappa }_4,{\xi }_4}+{\chi }_2)y\left(t\right)=f_2(t,x\left(t\right),y\left(t\right)),t\in \mathcal{J},}\hfill \end{array}\right.$$

(1)

equipped with the nonlocal integral boundary conditions

$$\left\{\begin{array}{c}{\displaystyle x\left(1\right)=0,x\left({\mu }_1\right)=\sum _{i=1}^{m_1}{\lambda }_i{}^{H}I_{1^{+}}^{{\alpha }_i}y\left({\eta }_i\right),x\left(T\right)=\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}y\left(s\right)ds,}\hfill \\ {\displaystyle y\left(1\right)=0,y\left({\mu }_2\right)=\sum _{l=1}^{m_2}{\sigma }_l{}^{H}I_{1^{+}}^{{\beta }_l}x\left({\nu }_l\right),y\left(T\right)=\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}x\left(s\right)ds,}\hfill \end{array}\right.$$

(2)

where ${}^{HH}D_{1^{+}}^{{\kappa }_p,{\xi }_p}$ denotes the Hilfer–Hadamard fractional derivative operator of order ${\kappa }_p$ and type ${\xi }_p$$(0\le {\xi }_p\le 1)$; $p=1,2,3,4,$ and ${}^{H}I_{1^{+}}^{{\alpha }_i}(i=1,\dots ,m_1)$ and ${}^{H}I_{1^{+}}^{{\beta }_l}(l=1,\dots ,m_2)$ are the Hadamard fractional integral operators of orders ${\alpha }_i,{\beta }_l>0$, respectively, with $1<{\kappa }_1,{\kappa }_3\le 2$ and $0<{\kappa }_2,{\kappa }_4\le 1$, ${\chi }_1,{\chi }_2>0$, ${\lambda }_i,{\sigma }_l,{\rho }_j,{\tau }_j\in \mathbb{R}$, $\mathcal{J}=[1,T]$, ${\mu }_1,{\mu }_2\in (1,T)$, ${\eta }_i,{\nu }_l\in (1,T)$, $1<{\zeta }_j<{\omega }_j<T$, $2<{\gamma }_1+{\kappa }_2\le 3$, $2<{\gamma }_2+{\kappa }_4\le 3$, and $f_1,f_2\in C(\mathcal{J}\times \mathbb{R}\times \mathbb{R},\mathbb{R})$.

We make use of the Leray–Schauder alternative and Banach’s fixed point theorem to derive results on the existence and uniqueness of the given problem, respectively. It is well known that the fixed point technique is an effective and fruitful method for developing a variety of results on the existence of boundary value problems under different criteria.

We have arranged the remaining content of this paper as follows. Some related concepts from fractional calculus and auxiliary results are presented in Section 2. Section 3 contains the main results, while examples illustrating these results are given in Section 4. We discuss the Ulam–Hyers stability for problem (1) and (2) in Section 5. New results for special cases are indicated in the final section.

2. Subsidiary Results

We begin this section with some definitions.

Definition 1

([38]). For a continuous function $g:[a,\infty )\to \mathbb{R}$, we define the Hadamard fractional integral of order $q>0$ as

$${}^{H}I_{a^{+}}^{q}g\left(t\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{a}{\overset{t}{\int }}{\left(log{\displaystyle \frac{t}{s}}\right)}^{q-1}{\displaystyle \frac{g\left(s\right)}{s}}ds,t>a,$$

where $log(.)={log}_e(.).$

Definition 2

([38]). For a continuous function $g:[a,\infty )\to \mathbb{R}$, the Hadamard fractional derivative of order $q>0$ is defined by

$${}^{H}D_{a^{+}}^{q}g\left(t\right)={\delta }^n\left({}^{H}I_{a+}^{n-q}g\right)\left(t\right),n=\left[q\right]+1,$$

where ${\delta }^n=t^n{\displaystyle \frac{d^n}{d{t}^n}}$, and $\left[q\right]$ denotes the integer part of the real number q.

Lemma 1

([38]). For $0<a<\infty ,$ and $p,q>0$, we have

• $\left({}^{H}I_{a+}^q{\left(log{\displaystyle \frac{t}{a}}\right)}^{p-1}\right)\left(x\right)={\displaystyle \frac{\Gamma \left(p\right)}{\Gamma (p+q)}}{\left(log{\displaystyle \frac{x}{a}}\right)}^{p+q-1};$
• $\left({}^{H}D_{a+}^q{\left(log{\displaystyle \frac{t}{a}}\right)}^{p-1}\right)\left(x\right)={\displaystyle \frac{\Gamma \left(p\right)}{\Gamma (p-q)}}{\left(log{\displaystyle \frac{x}{a}}\right)}^{p-q-1}.$

In particular, for $p=1$, we have

$$\left({}^{H}D_{a+}^{q}1\right)\left(x\right)={\displaystyle \frac{1}{\Gamma (1-q)}}{\left(log{\displaystyle \frac{x}{a}}\right)}^{-q}\ne 0,0<q<1.$$

Definition 3

([7]). We define the Hilfer–Hadamard fractional derivative of order $q\in (n-1,n]$ and type $p\in [0,1]$ for $g\in L^1(a,b),0<a<b<\infty$, as

$$\begin{array}{ccc}\hfill \left({}^{HH}D_{a+}^{q,p}g\right)\left(t\right)& =& \left({}^{H}I_{a+}^{p(n-q)}{\delta }^n{}^{H}I_{a+}^{(n-q)(1-p)}g\right)\left(t\right)\hfill \\ & & =\left({}^{H}I_{a+}^{p(n-q)}{\delta }^n{}^{H}I_{a+}^{(n-\gamma )}g\right)\left(t\right)\hfill \\ & & =\left({}^{H}I_{a+}^{p(n-q)}{}^{H}D_{a+}^{\gamma }g\right)\left(t\right),\gamma =q+np-qp,\hfill \end{array}$$

where ${}^{H}I_{a^{+}}^{(.)}$ and ${}^{H}D_{a^{+}}^{(.)}$ are given in Definitions 1 and 2, respectively.

Theorem 1

([39]). For $g\in L^1(a,b)$ and $\left({}^{H}I_{a+}^{n-\gamma }g\right)\left(t\right)\in A{C}_{\delta }^n[a,b]$, we have

$$\begin{array}{ccc}\hfill {\displaystyle {}^{H}I_{a^{+}}^q\left({}^{HH}D_{a+}^{q,p}g\right)\left(t\right)}& =& {}^{H}I_{a^{+}}^{\gamma }\left({}^{HH}D_{a+}^{\gamma }g\right)\left(t\right)\hfill \\ & & =g\left(t\right)-\sum _{j=0}^{n-1}{\displaystyle \frac{\left({\delta }^{(n-j-1)}\left({}^{H}I_{a^{+}}^{n-\gamma }g\right)\right)\left(a\right)}{\Gamma (\gamma -j)}}{\left(log{\displaystyle \frac{t}{a}}\right)}^{\gamma -j-1},\hfill \end{array}$$

where $0<a<b<\infty ,$ and $\gamma =q+np-qp$, $n=\left[q\right]+1,q>0,p\in [0,1]$ and $A{C}_{\delta }^n[a,b]=\{f:[a,b]\to \mathbb{R}:{\delta }^{n-1}\left[f\left(t\right)\right]\in AC[a,b]\}$. Notice that $\Gamma (\gamma -j)$ exists for all $j=1,2,\dots ,n-1$ and $\gamma \in [q,n]$.

In the current work, we write the Hadamard fractional integral operator ${}^{H}I_{a^{+}}^q$ and the Hilfer–Hadamard fractional derivative operator ${}^{HH}D_{a^{+}}^q$ as ${}^{H}I^q$ and ${}^{HH}D^q$, respectively.

In the following lemma, we solve the linear version of system (1) complemented with boundary data (2). This lemma plays a key role in transforming the given nonlinear problem into a fixed point problem.

Lemma 2.

For $h_1,h_2\in C(\mathcal{J},\mathbb{R})$, the unique solution of the linear system

$$\left\{\begin{array}{cc}{\displaystyle {}^{HH}D^{{\kappa }_1,{\xi }_1}({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)x\left(t\right)=h_1\left(t\right),}\hfill & t\in \mathcal{J},\\ {\displaystyle {}^{HH}D^{{\kappa }_3,{\xi }_3}({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)y\left(t\right)=h_2\left(t\right),}\hfill & t\in \mathcal{J},\end{array}\right.$$

(3)

subject to the boundary conditions in (2) is given by

$$\begin{array}{ccc}\hfill {\displaystyle x\left(t\right)}& =& {\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(u\right)}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(u\right)}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\},t\in \mathcal{J},\hfill \end{array}$$

(4)

and

$$\begin{array}{ccc}\hfill {\displaystyle y\left(t\right)}& =& {\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(u\right)}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(s\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{h_1\left(u\right)}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{h_2\left(s\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\},t\in \mathcal{J},\hfill \end{array}$$

(5)

where

$$\begin{array}{ccc}\hfill & & {\mathbb{M}}_1\left(t\right)=r_4{(logt)}^{{\gamma }_1+{\kappa }_2-1}+r_3{(logt)}^{{\gamma }_1+{\kappa }_2-2},{\mathbb{M}}_2\left(t\right)=z_4{(logt)}^{{\gamma }_1+{\kappa }_2-1}+z_3{(logt)}^{{\gamma }_1+{\kappa }_2-2},\hfill \\ & & {\mathbb{M}}_3\left(t\right)=u_4{(logt)}^{{\gamma }_1+{\kappa }_2-1}+u_3{(logt)}^{{\gamma }_1+{\kappa }_2-2},{\mathbb{M}}_4\left(t\right)=s_4{(logt)}^{{\gamma }_1+{\kappa }_2-1}+s_3{(logt)}^{{\gamma }_1+{\kappa }_2-2},\hfill \\ & & {\mathbb{N}}_1\left(t\right)=r_2{(logt)}^{{\gamma }_2+{\kappa }_4-1}+r_1{(logt)}^{{\gamma }_2+{\kappa }_4-2},{\mathbb{N}}_2\left(t\right)=z_2{(logt)}^{{\gamma }_2+{\kappa }_4-1}+z_1{(logt)}^{{\gamma }_2+{\kappa }_4-2},\hfill \\ & & {\mathbb{N}}_3\left(t\right)=u_2{(logt)}^{{\gamma }_2+{\kappa }_4-1}+u_1{(logt)}^{{\gamma }_2+{\kappa }_4-2},{\mathbb{N}}_4\left(t\right)=s_2{(logt)}^{{\gamma }_2+{\kappa }_4-1}+s_1{(logt)}^{{\gamma }_2+{\kappa }_4-2},\hfill \\ & & r_1={\displaystyle \frac{1}{A_1\Delta }}\left[-k_3\left(E_1+B_1{\Delta }_1\right)+k_1\left(L_1+B_1{\Delta }_2\right)\right],r_2={\displaystyle \frac{1}{A_1{k}_1}}\left[E_1-A_1{k}_2{r}_1+B_1{\Delta }_1\right],\hfill \\ & & r_3=-{\displaystyle \frac{1}{{\Delta }_3}}\left[B_1+\left(A_4{B}_1-B_4{A}_1\right)r_1+\left(A_3{B}_1-B_3{A}_1\right)r_2\right],r_4={\displaystyle \frac{1}{A_1}}\left(-A_2{r}_3+A_4{r}_1+A_3{r}_2+1\right),\hfill \\ & & z_1={\displaystyle \frac{1}{\Delta }}\left(k_3{\Delta }_1-k_1{\Delta }_2\right),z_2={\displaystyle \frac{-1}{k_1}}\left({\Delta }_1+k_2{z}_1\right),z_3={\displaystyle \frac{1}{{\Delta }_3}}\left[A_1-\left(A_4{B}_1-B_4{A}_1\right)z_1-\left(A_3{B}_1-B_3{A}_1\right)z_2\right],\hfill \\ & & z_4={\displaystyle \frac{-1}{A_1}}\left(A_2{z}_3-A_3{z}_2-A_4{z}_1\right),u_1={\displaystyle \frac{-k_3}{\Delta }},u_2={\displaystyle \frac{1}{k_1}}\left(1-k_2{u}_1\right),\hfill \\ & & u_3=-{\displaystyle \frac{1}{{\Delta }_3}}\left[\left(A_3{B}_1-B_3{A}_1\right)u_2+\left(A_4{B}_1-B_4{A}_1\right)u_1\right],u_4={\displaystyle \frac{1}{A_1}}\left[-A_2{u}_3+A_3{u}_2+A_4{u}_1\right],s_1={\displaystyle \frac{k_1}{\Delta }},\hfill \\ & & s_2={\displaystyle \frac{-k_2}{\Delta }},s_3=-{\displaystyle \frac{1}{\Delta {\Delta }_3}}\left[k_1\left(A_4{B}_1-B_4{A}_1\right)-k_2\left(A_3{B}_1-B_3{A}_1\right)\right],s_4={\displaystyle \frac{1}{A_1}}\left(A_4{s}_1-A_2{s}_3+A_3{s}_2\right),\hfill \\ & & {\Delta }_1={\displaystyle \frac{1}{{\Delta }_3}}(A_2{E}_1-E_2{A}_1),{\Delta }_2={\displaystyle \frac{1}{{\Delta }_3}}(A_2{L}_1-L_2{A}_1),{\Delta }_3=A_1{B}_2-B_1{A}_2\ne 0,\Delta =k_1{k}_4-k_2{k}_3\ne 0,\hfill \\ & & k_1={\displaystyle \frac{1}{A_1}}\left[(A_1{E}_3-E_1{A}_3)-{\Delta }_1(A_3{B}_1-B_3{A}_1)\right],k_2={\displaystyle \frac{1}{A_1}}\left[(A_1{E}_4-E_1{A}_4)-{\Delta }_1(A_4{B}_1-B_4{A}_1)\right],\hfill \\ & & k_3={\displaystyle \frac{1}{A_1}}\left[(A_1{L}_3-L_1{A}_3)-{\Delta }_2(A_3{B}_1-B_3{A}_1)\right],k_4={\displaystyle \frac{1}{A_1}}\left[(A_1{L}_4-L_1{A}_4)-{\Delta }_2(A_4{B}_1-B_4{A}_1)\right],\hfill \\ & & A_1={(log{\mu }_1)}^{{\gamma }_1+{\kappa }_2-1},A_2={(log{\mu }_1)}^{{\gamma }_1+{\kappa }_2-2},A_3=\sum _{i=1}^{m_1}{\displaystyle \frac{{\lambda }_i\Gamma ({\gamma }_2+{\kappa }_4)}{\Gamma ({\alpha }_i+{\gamma }_2+{\kappa }_4)}}{(log{\eta }_i)}^{{\alpha }_i+{\gamma }_2+{\kappa }_4-1},\hfill \\ & & A_4=\sum _{i=1}^{m_1}{\displaystyle \frac{{\lambda }_i\Gamma ({\gamma }_2+{\kappa }_4-1)}{\Gamma ({\alpha }_i+{\gamma }_2+{\kappa }_4-1)}}{(log{\eta }_i)}^{{\alpha }_i+{\gamma }_2+{\kappa }_4-2},B_1={(logT)}^{{\gamma }_1+{\kappa }_2-1},B_2={(logT)}^{{\gamma }_1+{\kappa }_2-2},\hfill \\ & & B_3={\displaystyle \frac{1}{({\gamma }_2+{\kappa }_4)}}\sum _{j=1}^n{\rho }_j\left[{(log{\omega }_j)}^{{\gamma }_2+{\kappa }_4}-{(log{\zeta }_j)}^{{\gamma }_2+{\kappa }_4}\right],\hfill \\ & & B_4={\displaystyle \frac{1}{({\gamma }_2+{\kappa }_4-1)}}\sum _{j=1}^n{\rho }_j\left[{(log{\omega }_j)}^{{\gamma }_2+{\kappa }_4-1}-{(log{\zeta }_j)}^{{\gamma }_2+{\kappa }_4-1}\right],\hfill \\ & & E_1=\sum _{l=1}^{m_2}{\displaystyle \frac{{\sigma }_l\Gamma ({\gamma }_1+{\kappa }_2)}{\Gamma ({\beta }_l+{\gamma }_1+{\kappa }_2)}}{(log{\nu }_l)}^{{\beta }_l+{\gamma }_1+{\kappa }_2-1},E_2=\sum _{l=1}^{m_2}{\displaystyle \frac{{\sigma }_l\Gamma ({\gamma }_1+{\kappa }_2-1)}{\Gamma ({\beta }_l+{\gamma }_1+{\kappa }_2-1)}}{(log{\nu }_l)}^{{\beta }_l+{\gamma }_1+{\kappa }_2-2},\hfill \\ & & E_3={(log{\mu }_2)}^{{\gamma }_2+{\kappa }_4-1},E_4={(log{\mu }_2)}^{{\gamma }_2+{\kappa }_4-2},L_1={\displaystyle \frac{1}{({\gamma }_1+{\kappa }_2)}}\sum _{j=1}^n{\tau }_j\left[{(log{\omega }_j)}^{{\gamma }_1+{\kappa }_2}-{(log{\zeta }_j)}^{{\gamma }_1+{\kappa }_2}\right],\hfill \\ & & L_2={\displaystyle \frac{1}{({\gamma }_1+{\kappa }_2-1)}}\sum _{j=1}^n{\tau }_j\left[{(log{\omega }_j)}^{{\gamma }_1+{\kappa }_2-1}-{(log{\zeta }_j)}^{{\gamma }_1+{\kappa }_2-1}\right],\hfill \\ & & L_3={(logT)}^{{\gamma }_2+{\kappa }_4-1},L_4={(logT)}^{{\gamma }_2+{\kappa }_4-2}.\hfill \end{array}$$

(6)

Proof. 

Using the Hadamard integral operators ${}^{H}I^{{\kappa }_1}$ and ${}^{H}I^{{\kappa }_3}$ on the first and second equations in (3), respectively, we obtain

$$\left\{\begin{array}{c}{\displaystyle ({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)x\left(t\right)={}^{H}I^{{\kappa }_1}{h}_1\left(t\right)+c_1{(logt)}^{{\gamma }_1-1}+c_2{(logt)}^{{\gamma }_1-2},t\in \mathcal{J},}\hfill \\ {\displaystyle ({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)y\left(t\right)={}^{H}I^{{\kappa }_3}{h}_2\left(t\right)+d_1{(logt)}^{{\gamma }_2-1}+d_2{(logt)}^{{\gamma }_2-2},t\in \mathcal{J},}\hfill \end{array}\right.$$

(7)

where ${\gamma }_1={\kappa }_1+(2-{\kappa }_1){\xi }_1,$${\gamma }_2={\kappa }_3+(2-{\kappa }_3){\xi }_3$, and $c_1,c_2,d_1,d_2\in \mathbb{R}$ are unknown arbitrary constants.

Now, applying the Hadamard integral operators ${}^{H}I^{{\kappa }_2}$ and ${}^{H}I^{{\kappa }_4}$ to the first and second equations in (7), respectively, we obtain

$$\left\{\begin{array}{cc}x\left(t\right)\hfill & {\displaystyle ={}^{H}I^{{\kappa }_1+{\kappa }_2}{h}_1\left(t\right)-{\chi }_1{}^{H}I^{{\kappa }_2}x\left(t\right)+c_1\frac{\Gamma \left({\gamma }_1\right)}{\Gamma ({\gamma }_1+{\kappa }_2)}{(logt)}^{{\gamma }_1+{\kappa }_2-1}}\hfill \\ \hfill & {\displaystyle +c_2\frac{\Gamma ({\gamma }_1-1)}{\Gamma ({\gamma }_1+{\kappa }_2-1)}{(logt)}^{{\gamma }_1+{\kappa }_2-2}+c_3{(logt)}^{{\gamma }_3-1},t\in \mathcal{J},}\hfill \\ {\displaystyle y\left(t\right)}\hfill & {\displaystyle ={}^{H}I^{{\kappa }_3+{\kappa }_4}{h}_2\left(t\right)-{\chi }_2{}^{H}I^{{\kappa }_4}y\left(t\right)+d_1\frac{\Gamma \left({\gamma }_2\right)}{\Gamma ({\gamma }_2+{\kappa }_4)}{(logt)}^{{\gamma }_2+{\kappa }_4-1}}\hfill \\ \hfill & {\displaystyle +d_2\frac{\Gamma ({\gamma }_2-1)}{\Gamma ({\gamma }_2+{\kappa }_4-1)}{(logt)}^{{\gamma }_2+{\kappa }_4-2}+d_3{(logt)}^{{\gamma }_4-1},t\in \mathcal{J},}\hfill \end{array}\right.$$

(8)

where ${\gamma }_3={\kappa }_2+(1-{\kappa }_2){\xi }_2,{\gamma }_4={\kappa }_4+(1-{\kappa }_4){\xi }_4$, and $c_3,d_3\in \mathbb{R}$ are unknown arbitrary constants.

Combining the conditions $x\left(1\right)=0=y\left(1\right)$ with (8), we find that $c_3=d_3=0$ as ${\gamma }_3\in [{\kappa }_2,1]$ and ${\gamma }_4\in [{\kappa }_4,1].$ Letting ${\tilde{c}}_1=c_1{\displaystyle \frac{\Gamma \left({\gamma }_1\right)}{\Gamma ({\gamma }_1+{\kappa }_2)}},$${\tilde{c}}_2=c_2{\displaystyle \frac{\Gamma ({\gamma }_1-1)}{\Gamma ({\gamma }_1+{\kappa }_2-1)}},$${\tilde{d}}_1=d_1{\displaystyle \frac{\Gamma \left({\gamma }_2\right)}{\Gamma ({\gamma }_2+{\kappa }_4)}},$ and ${\tilde{d}}_2=d_2{\displaystyle \frac{\Gamma ({\gamma }_2-1)}{\Gamma ({\gamma }_2+{\kappa }_4-1)}},$ system (8) together with $c_3=d_3=0$ takes the form

$$\left\{\begin{array}{cc}{\displaystyle x\left(t\right)}\hfill & {\displaystyle ={}^{H}I^{{\kappa }_1+{\kappa }_2}{h}_1\left(t\right)-{\chi }_1{}^{H}I^{{\kappa }_2}x\left(t\right)+{\tilde{c}}_1{(logt)}^{{\gamma }_1+{\kappa }_2-1}+{\tilde{c}}_2{(logt)}^{{\gamma }_1+{\kappa }_2-2},t\in \mathcal{J},}\hfill \\ {\displaystyle y\left(t\right)}\hfill & {\displaystyle ={}^{H}I^{{\kappa }_3+{\kappa }_4}{h}_2\left(t\right)-{\chi }_2{}^{H}I^{{\kappa }_4}y\left(t\right)+{\tilde{d}}_1{(logt)}^{{\gamma }_2+{\kappa }_4-1}+{\tilde{d}}_2{(logt)}^{{\gamma }_2+{\kappa }_4-2},t\in \mathcal{J}.}\hfill \end{array}\right.$$

(9)

Now, using (9) for the remaining conditions given by (2), we obtain

$$\begin{array}{cc}\hfill A_1{\tilde{c}}_1+A_2{\tilde{c}}_2-A_3{\tilde{d}}_1-A_4{\tilde{d}}_2& ={\mathbb{J}}_1,\\ \hfill B_1{\tilde{c}}_1+B_2{\tilde{c}}_2-B_3{\tilde{d}}_1-B_4{\tilde{d}}_2& ={\mathbb{J}}_2,\\ \hfill -E_1{\tilde{c}}_1-E_2{\tilde{c}}_2+E_3{\tilde{d}}_1+E_4{\tilde{d}}_2& ={\mathbb{J}}_3,\\ \hfill -L_1{\tilde{c}}_1-L_2{\tilde{c}}_2+L_3{\tilde{d}}_1+L_4{\tilde{d}}_2& ={\mathbb{J}}_4,\end{array}$$

(10)

where $A_p,B_p,E_p,L_p,p=1,2,3,4,$ are given in (6), and

$$\begin{array}{ccc}\hfill & & {\mathbb{J}}_1=\sum _{i=1}^{m_1}{\lambda }_i\left[{}^{H}I^{{\alpha }_i+{\kappa }_3+{\kappa }_4}{h}_2\left({\eta }_i\right)-{\chi }_2{}^{H}I^{{\alpha }_i+{\kappa }_4}y\left({\eta }_i\right)\right]-\left[{}^{H}I^{{\kappa }_1+{\kappa }_2}{h}_1\left({\mu }_1\right)-{\chi }_1{}^{H}I^{\kappa 2}x\left({\mu }_1\right)\right],\hfill \\ & & {\mathbb{J}}_2=\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}\left[{}^{H}I^{{\kappa }_3+{\kappa }_4}{h}_2\left(s\right)-{\chi }_2{}^{H}I^{{\kappa }_4}y\left(s\right)\right]ds-\left[{}^{H}I^{{\kappa }_1+{\kappa }_2}{h}_1\left(T\right)-{\chi }_1{}^{H}I^{{\kappa }_2}x\left(T\right)\right],\hfill \\ & & {\mathbb{J}}_3=\sum _{l=1}^{m_2}{\sigma }_l\left[{}^{H}I^{{\beta }_l+{\kappa }_1+{\kappa }_2}{h}_1\left({\nu }_l\right)-{\chi }_1{}^{H}I^{{\beta }_l+{\kappa }_2}x\left({\nu }_l\right)\right]-\left[{}^{H}I^{{\kappa }_3+{\kappa }_4}{h}_2\left({\mu }_2\right)-{\chi }_2{}^{H}I^{\kappa 4}y\left({\mu }_2\right)\right],\hfill \\ & & {\mathbb{J}}_4=\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}\left[{}^{H}I^{{\kappa }_1+{\kappa }_2}{h}_1\left(s\right)-{\chi }_1{}^{H}I^{{\kappa }_2}x\left(s\right)\right]ds-\left[{}^{H}I^{{\kappa }_3+{\kappa }_4}{h}_2\left(T\right)-{\chi }_2{}^{H}I^{{\kappa }_4}y\left(T\right)\right].\hfill \end{array}$$

(11)

Solving system (10) for ${\tilde{c}}_1,{\tilde{c}}_2,{\tilde{d}}_1$, and ${\tilde{d}}_2$, we find that

$$\begin{array}{ccc}\hfill & & {\tilde{c}}_1=r_4{\mathbb{J}}_1+z_4{\mathbb{J}}_2+u_4{\mathbb{J}}_3+s_4{\mathbb{J}}_4,\hfill \\ & & {\tilde{c}}_2=r_3{\mathbb{J}}_1+z_3{\mathbb{J}}_2+u_3{\mathbb{J}}_3+s_3{\mathbb{J}}_4,\hfill \\ & & {\tilde{d}}_1=r_2{\mathbb{J}}_1+z_2{\mathbb{J}}_2+u_2{\mathbb{J}}_3+s_2{\mathbb{J}}_4,\hfill \\ & & {\tilde{d}}_2=r_1{\mathbb{J}}_1+z_1{\mathbb{J}}_2+u_1{\mathbb{J}}_3+s_1{\mathbb{J}}_4.\hfill \end{array}$$

(12)

Using (12) together with (6) and (11) in (9), we obtain the solution (4) and (5). The converse of the lemma follows through direct computation. □

3. The Main Results

According to Lemma 2, problem(1) and (2) can be transformed into a fixed point problem as $(x,y)=\mathbb{A}(x,y)$, where $\mathbb{A}:\mathbb{X}\times \mathbb{X}\to \mathbb{X}\times \mathbb{X}$ is an operator defined by

$$\mathbb{A}(x,y)\left(t\right)=\left(\begin{array}{c}{\mathbb{A}}_1(x,y)\left(t\right)\\ {\mathbb{A}}_2(x,y)\left(t\right)\end{array}\right),$$

(13)

with

$$\begin{array}{ccc}\hfill {\mathbb{A}}_1(x,y)\left(t\right)& =& {\displaystyle {\int }_1^t\left[\frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{f_1(s,x\left(s\right),y\left(s\right))}{s}-{\chi }_1\frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{x\left(s\right)}{s}\right]ds}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(u,x\left(u\right),y\left(u\right))}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(u,x\left(u\right),y\left(u\right))}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\},\hfill \end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill {\mathbb{A}}_2(x,y)\left(t\right)& =& {\displaystyle {\int }_1^t\left[\frac{{(log\frac{t}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{f_2(s,x\left(s\right),y\left(s\right))}{s}-{\chi }_2\frac{{(log\frac{t}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{y\left(s\right)}{s}\right]ds}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(u,x\left(u\right),y\left(u\right))}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{N}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(u,x\left(u\right),y\left(u\right))}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{y\left(s\right)}{s}}\right]ds\}.\hfill \end{array}$$

(15)

Here, the product space $\mathbb{X}\times \mathbb{X}$ is a Banach space equipped with the norm $\parallel (x,y)\parallel =\parallel x\parallel +\parallel y\parallel ,(x,y)\in \mathbb{X}\times \mathbb{X}$, where $\mathbb{X}$ is the Banach space of all of the continuous functions from $\mathcal{J}\to \mathbb{R}$ endowed with the supremum norm $\parallel x\parallel ={sup}_{t\in \mathcal{J}}\left|x\left(t\right)\right|.$

Note that the fixed points of the operator $\mathbb{A}$ correspond to solutions to problem (1) and (2).

In the sequel, we need the following assumptions:

$\left(H_1\right)$

Real constants ${\widehat{m}}_p,{\widehat{n}}_p\ge 0,p=1,2$, and ${\widehat{m}}_0,{\widehat{n}}_0>0$ exist such that $\forall x,y\in \mathbb{R},$

$$\begin{array}{c}\hfill |f_1(t,x,y)|\le {\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|,\\ \hfill |f_2(t,x,y)|\le {\widehat{n}}_0+{\widehat{n}}_1\left|x\right|+{\widehat{n}}_2\left|y\right|;\end{array}$$

$\left(H_2\right)$

For all $t\in \mathcal{J},x_p,y_p\in \mathbb{R},p=1,2$, positive constants exist ${\mathbb{L}}_1$ and ${\mathbb{L}}_2$ such that

$$|f_1(t,x_2,y_2)-f_1(t,x_1,y_1|\le {\mathbb{L}}_1\left(\right|x_2-x_1|+|y_2-y_1\left|\right),$$

$$|f_2(t,x_2,y_2)-f_2(t,x_1,y_1|\le {\mathbb{L}}_2\left(\right|x_2-x_1|+|y_2-y_1\left|\right).$$

Next, we introduce the notation

$$\begin{array}{ccc}\hfill {\displaystyle {\Psi }_1}& =& {\displaystyle \frac{{(logT)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}+{\overline{\mathbb{M}}}_1{\displaystyle \frac{{(log{\mu }_1)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}+{\overline{\mathbb{M}}}_2{\displaystyle \frac{{(logT)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}+{\overline{\mathbb{M}}}_3\sum _{l=1}^{m_2}{\displaystyle \frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_1+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2+1)}}\hfill \\ & & +{\overline{\mathbb{M}}}_4\sum _{j=1}^n{\displaystyle \frac{|{\tau }_j|}{\Gamma ({\kappa }_1+{\kappa }_2+2)}}\left({(log{\omega }_j)}^{{\kappa }_1+{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_1+{\kappa }_2+1}\right),\hfill \\ \hfill {\displaystyle {\Psi }_2}& =& {\overline{\mathbb{M}}}_1\sum _{i=1}^{m_1}{\displaystyle \frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_3+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4+1)}}+{\overline{\mathbb{M}}}_2\sum _{j=1}^n{\displaystyle \frac{|{\rho }_j|}{\Gamma ({\kappa }_3+{\kappa }_4+2)}}\left({(log{\omega }_j)}^{{\kappa }_3+{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_3+{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{M}}}_3{\displaystyle \frac{{(log{\mu }_2)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}}+{\overline{\mathbb{M}}}_4{\displaystyle \frac{{(logT)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}},\hfill \\ \hfill {\displaystyle {\Psi }_3}& =& {\chi }_1[{\displaystyle \frac{{(logT)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}}+{\overline{\mathbb{M}}}_1{\displaystyle \frac{{(log{\mu }_1)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}}+{\overline{\mathbb{M}}}_2{\displaystyle \frac{{(logT)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}}+{\overline{\mathbb{M}}}_3\sum _{l=1}^{m_2}{\displaystyle \frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_2+1)}}\hfill \\ & & +{\overline{\mathbb{M}}}_4\sum _{j=1}^n{\displaystyle \frac{|{\tau }_j|}{\Gamma ({\kappa }_2+2)}}\left({(log{\omega }_j)}^{{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_2+1}\right)],\hfill \\ \hfill {\displaystyle {\Psi }_4}& =& {\chi }_2[{\overline{\mathbb{M}}}_1\sum _{i=1}^{m_1}{\displaystyle \frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_4+1)}}+{\overline{\mathbb{M}}}_2\sum _{j=1}^n{\displaystyle \frac{|{\rho }_j|}{\Gamma ({\kappa }_4+2)}}\left({(log{\omega }_j)}^{{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{M}}}_3{\displaystyle \frac{{(log{\mu }_2)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}}+{\overline{\mathbb{M}}}_4{\displaystyle \frac{{(logT)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}}],\hfill \\ \hfill {\displaystyle {\Phi }_1}& =& {\overline{\mathbb{N}}}_1{\displaystyle \frac{{(log{\mu }_1)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}+{\overline{\mathbb{N}}}_2{\displaystyle \frac{{(logT)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}+{\overline{\mathbb{N}}}_3\sum _{l=1}^{m_2}{\displaystyle \frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_1+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2+1)}}\hfill \\ & & +{\overline{\mathbb{N}}}_4\sum _{j=1}^n{\displaystyle \frac{|{\tau }_j|}{\Gamma ({\kappa }_1+{\kappa }_2+2)}}\left({(log{\omega }_j)}^{{\kappa }_1+{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_1+{\kappa }_2+1}\right),\hfill \\ \hfill {\Phi }_2& =& {\displaystyle \frac{{(logT)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}}+{\overline{\mathbb{N}}}_1\sum _{i=1}^{m_1}{\displaystyle \frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_3+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4+1)}}\hfill \\ & & +{\overline{\mathbb{N}}}_2\sum _{j=1}^n{\displaystyle \frac{|{\rho }_j|}{\Gamma ({\kappa }_3+{\kappa }_4+2)}}\left({(log{\omega }_j)}^{{\kappa }_3+{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_3+{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{N}}}_3{\displaystyle \frac{{(log{\mu }_2)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}}+{\overline{\mathbb{N}}}_4{\displaystyle \frac{{(logT)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}},\hfill \\ \hfill {\Phi }_3& =& {\chi }_1[{\overline{\mathbb{N}}}_1{\displaystyle \frac{{(log{\mu }_1)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}}+{\overline{\mathbb{N}}}_2{\displaystyle \frac{{(logT)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}}+{\overline{\mathbb{N}}}_3\sum _{l=1}^{m_2}{\displaystyle \frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_2+1)}}\hfill \\ & & +{\overline{\mathbb{N}}}_4\sum _{j=1}^n{\displaystyle \frac{|{\tau }_j|}{\Gamma ({\kappa }_2+2)}}\left({(log{\omega }_j)}^{{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_2+1}\right)],\hfill \\ \hfill {\Phi }_4& =& {\chi }_2[{\displaystyle \frac{{(logT)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}}+{\overline{\mathbb{N}}}_1\sum _{i=1}^{m_1}{\displaystyle \frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_4+1)}}+{\overline{\mathbb{N}}}_2\sum _{j=1}^n{\displaystyle \frac{|{\rho }_j|}{\Gamma ({\kappa }_4+2)}}\left({(log{\omega }_j)}^{{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{N}}}_3{\displaystyle \frac{{(log{\mu }_2)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}}+{\overline{\mathbb{N}}}_4{\displaystyle \frac{{(logT)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}}],\hfill \end{array}$$

(16)

where

$${\overline{\mathbb{M}}}_p=\underset{t\in \mathcal{J}}{sup}|{\mathbb{M}}_p\left(t\right)|,{\overline{\mathbb{N}}}_p=\underset{t\in \mathcal{J}}{sup}{|\mathbb{N}}_p\left(t\right)|,p=1,2,3,4.$$

Now, we proceed to presenting our main results. In our first result, we establish the existence of solutions to the problem(1) and (2), which relies on the Leray–Schauder alternative [40].

Theorem 2.

Let $f_1,f_2\in C(\mathcal{J}\times \mathbb{R}\times \mathbb{R},\mathbb{R})$, and the condition ($H_1$) is satisfied. Then, at least one solution to the problem (1) and (2) on $\mathcal{J}$existsif $0<{\mathbb{S}}_1,{\mathbb{S}}_2<1$, where

$$\begin{array}{ccc}\hfill & & {\mathbb{S}}_1={\widehat{m}}_1({\Psi }_1+{\Phi }_1)+{\widehat{n}}_1({\Psi }_2+{\Phi }_2)+({\Psi }_3+{\Phi }_3),\hfill \\ & & {\mathbb{S}}_2={\widehat{m}}_2({\Psi }_1+{\Phi }_1)+{\widehat{n}}_2({\Psi }_2+{\Phi }_2)+({\Psi }_4+{\Phi }_4),\hfill \end{array}$$

(17)

and ${\Psi }_p$, ${\Phi }_p$, $p=1,2,3,4,$ are given in (16).

Proof. 

Let us first establish that the operator $\mathbb{A}:\mathbb{X}\times \mathbb{X}\to \mathbb{X}\times \mathbb{X}$, given by (13), is completely continuous. Obviously, the operator $\mathbb{A}$ is continuous in view of the continuity of $f_1$ and $f_2.$ For a fixed number $\widehat{r}$, we consider a bounded set $\mathbb{O}=\{(x,y)\in \mathbb{X}\times \mathbb{X}:\parallel (x,y)\parallel \le \widehat{r}\}\subset \mathbb{X}\times \mathbb{X}.$ According to ($H_1$), we have

$$\begin{array}{ccc}\hfill |f_1(t,x,y)|& \le & {\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\hfill \\ & \le & {\widehat{m}}_0+{\widehat{m}}_1(\parallel x\parallel +\parallel y\parallel )+{\widehat{m}}_2(\parallel x\parallel +\parallel y\parallel )\hfill \\ & \le & {\widehat{m}}_0+({\widehat{m}}_1+{\widehat{m}}_2)\widehat{r}=:{\mathbb{K}}_1.\hfill \end{array}$$

Likewise, we have $|f_2(t,x,y)|\le {\widehat{n}}_0+({\widehat{n}}_1+{\widehat{n}}_2)\widehat{r}=:{\mathbb{K}}_2.$ Then, for any $(x,y)\in \mathbb{O}$, we obtain

$$\begin{array}{ccc}\hfill & & \parallel {\mathbb{A}}_1(x,y)∥\hfill \\ & \le & \underset{t\in \mathcal{J}}{sup}\{{\int }_1^t\left[\frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_1\frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +|{\mathbb{M}}_1\left(t\right)\left|\right\{\sum _{i=1}^{m_1}\left|{\lambda }_i\right|{\int }_1^{{\eta }_i}\left[\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_2\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}\frac{\left|y\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_1}\left[\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_1\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_2\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\rho }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(u,x\left(u\right),y\left(u\right))|}{u}+{\chi }_2\frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(u\left)\right|}{u}\right]duds\hfill \\ \hfill & & +{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_1\frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_3\left(t\right)\left|\right\{\sum _{l=1}^{m_2}\left|{\sigma }_l\right|{\int }_1^{{\nu }_l}\left[\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_1\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_2}\left[\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_2\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_4\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\tau }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(u,x\left(u\right),y\left(u\right))|}{u}+{\chi }_1\frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(u\left)\right|}{u}\right]duds\hfill \\ \hfill & & +{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x\left(s\right),y\left(s\right))|}{s}+{\chi }_2\frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(s\left)\right|}{s}\right]ds\left\}\right\}\hfill \\ & \le & {\mathbb{K}}_1[\frac{{(logT)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}+{\overline{\mathbb{M}}}_1\frac{{(log{\mu }_1)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}+{\overline{\mathbb{M}}}_2\frac{{(logT)}^{{\kappa }_1+{\kappa }_2}}{\Gamma ({\kappa }_1+{\kappa }_2+1)}+{\overline{\mathbb{M}}}_3\sum _{l=1}^{m_2}\frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_1+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2+1)}\hfill \\ & & +{\overline{\mathbb{M}}}_4\sum _{j=1}^n\frac{|{\tau }_j|}{\Gamma ({\kappa }_1+{\kappa }_2+2)}\left({(log{\omega }_j)}^{{\kappa }_1+{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_1+{\kappa }_2+1}\right)]\hfill \\ & & +{\mathbb{K}}_2[{\overline{\mathbb{M}}}_1\sum _{i=1}^{m_1}\frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_3+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4+1)}+{\overline{\mathbb{M}}}_2\sum _{j=1}^n\frac{|{\rho }_j|}{\Gamma ({\kappa }_3+{\kappa }_4+2)}\left({(log{\omega }_j)}^{{\kappa }_3+{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_3+{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{M}}}_3\frac{{(log{\mu }_2)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}+{\overline{\mathbb{M}}}_4\frac{{(logT)}^{{\kappa }_3+{\kappa }_4}}{\Gamma ({\kappa }_3+{\kappa }_4+1)}]\hfill \\ & & +{\chi }_1[\frac{{(logT)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}+{\overline{\mathbb{M}}}_1\frac{{(log{\mu }_1)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}+{\overline{\mathbb{M}}}_2\frac{{(logT)}^{{\kappa }_2}}{\Gamma ({\kappa }_2+1)}+{\overline{\mathbb{M}}}_3\sum _{l=1}^{m_2}\frac{|{\sigma }_l|{(log{\nu }_l)}^{{\beta }_l+{\kappa }_2}}{\Gamma ({\beta }_l+{\kappa }_2+1)}\hfill \\ & & +{\overline{\mathbb{M}}}_4\sum _{j=1}^n\frac{|{\tau }_j|}{\Gamma ({\kappa }_2+2)}\left({(log{\omega }_j)}^{{\kappa }_2+1}-{(log{\zeta }_j)}^{{\kappa }_2+1}\right)]\parallel x\parallel \hfill \\ & & +{\chi }_2[{\overline{\mathbb{M}}}_1\sum _{i=1}^{m_1}\frac{|{\lambda }_i|{(log{\eta }_i)}^{{\alpha }_i+{\kappa }_4}}{\Gamma ({\alpha }_i+{\kappa }_4+1)}+{\overline{\mathbb{M}}}_2\sum _{j=1}^n\frac{|{\rho }_j|}{\Gamma ({\kappa }_4+2)}\left({(log{\omega }_j)}^{{\kappa }_4+1}-{(log{\zeta }_j)}^{{\kappa }_4+1}\right)\hfill \\ & & +{\overline{\mathbb{M}}}_3\frac{{(log{\mu }_2)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}+{\overline{\mathbb{M}}}_4\frac{{(logT)}^{{\kappa }_4}}{\Gamma ({\kappa }_4+1)}]\parallel y\parallel \hfill \\ \hfill & & \\ & \le & {\mathbb{K}}_1{\Psi }_1+{\mathbb{K}}_2{\Psi }_2+{\Psi }_3\parallel x\parallel +{\Psi }_4\parallel y\parallel ,\hfill \end{array}$$

(18)

where ${\Psi }_p,p=1,2,3,4,$ are given in (16).

Similarly, one can find that

$$\parallel {\mathbb{A}}_2(x,y)\parallel \le {\mathbb{K}}_1{\Phi }_1+{\mathbb{K}}_2{\Phi }_2+{\Phi }_3\parallel x\parallel +{\Phi }_4\parallel y\parallel ,$$

(19)

where ${\Phi }_p,p=1,2,3,4,$ are given in (16).

From (18) and (19), we have

$$\begin{array}{ccc}\hfill \parallel \mathbb{A}(x,y)\parallel & =& \parallel {\mathbb{A}}_1(x,y)\parallel +\parallel {\mathbb{A}}_2(x,y)\parallel \hfill \\ & \le & {\mathbb{K}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{K}}_2({\Psi }_2+{\Phi }_2)+max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)\parallel (x,y)\parallel \hfill \\ & \le & {\mathbb{K}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{K}}_2({\Psi }_2+{\Phi }_2)+max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)\widehat{r},\hfill \end{array}$$

which implies that $\mathbb{A}\left(\mathbb{O}\right)$ is uniformly bounded.

To show that $\mathbb{A}\left(\mathbb{O}\right)$ is equicontinuous, we take $t_1,t_2\in \mathcal{J}$ with $t_1<t_2.$ Then, we obtain

$$\begin{array}{ccc}& & |{\mathbb{A}}_1(x,y)\left(t_2\right)-{\mathbb{A}}_1(x,y)\left(t_1\right)|\hfill \\ & \le & |{\int }_1^{t_2}\left[{\displaystyle \frac{{(log\frac{t_2}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t_2}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ & & -{\int }_1^{t_2}\left[{\displaystyle \frac{{(log\frac{t_2}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t_2}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]\hfill \\ & & -[{\int }_1^{t_1}\left[{\displaystyle \frac{{(log\frac{t_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]ds\hfill \\ & & -{\int }_1^{t_1}\left[{\displaystyle \frac{{(log\frac{t_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,x\left(s\right),y\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{x\left(s\right)}{s}}\right]\left]\right|\hfill \\ & & +|{\mathbb{M}}_1\left(t_2\right)-{\mathbb{M}}_1\left(t_2\right)\left|\right|{\mathbb{J}}_1|+|{\mathbb{M}}_2\left(t_2\right)-{\mathbb{M}}_2\left(t_2\right)\left|\right|{\mathbb{J}}_2|\hfill \\ & & +|{\mathbb{M}}_3\left(t_2\right)-{\mathbb{M}}_3\left(t_2\right)\left|\right|{\mathbb{J}}_3|+|{\mathbb{M}}_4\left(t_2\right)-{\mathbb{M}}_4\left(t_2\right)\left|\right|{\mathbb{J}}_4|\hfill \\ & \le & {\displaystyle \frac{{\mathbb{K}}_1}{\Gamma ({\kappa }_1+{\kappa }_2+1)}}\left[2{\left(log{t}_2-log{t}_1\right)}^{{\kappa }_1+{\kappa }_2}+|{(log{t}_2)}^{{\kappa }_1+{\kappa }_2}-{(log{t}_1)}^{{\kappa }_1+{\kappa }_2}|\right]\hfill \\ & & +{\displaystyle \frac{{\chi }_1}{\Gamma ({\kappa }_2+1)}}\left[2{\left(log{t}_2-log{t}_1\right)}^{{\kappa }_2}+|{(log{t}_2)}^{{\kappa }_2}-{(log{t}_1)}^{{\kappa }_2}|\right]\widehat{r}\hfill \\ & & +\left[\right|r_4\left|\right|{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-1}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-1}|\hfill \\ & & +\left|r_3\right||{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-2}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-2}|\left]\right|{\mathbb{J}}_1|\hfill \\ & & +\left[\right|z_4\left|\right|{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-1}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-1}|\hfill \\ & & +\left|z_3\right||{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-2}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-2}|\left]\right|{\mathbb{J}}_2|\hfill \\ & & +\left[\right|u_4\left|\right|{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-1}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-1}|\hfill \\ & & +\left|u_3\right||{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-2}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-2}|\left]\right|{\mathbb{J}}_3|\hfill \\ & & +\left[\right|s_4\left|\right|{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-1}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-1}|\hfill \\ & & +\left|s_3\right||{(log{t}_2)}^{{\gamma }_1+{\kappa }_2-2}-{(log{t}_1)}^{{\gamma }_1+{\kappa }_2-2}|\left]\right|{\mathbb{J}}_4|\hfill \\ \hfill \\ & ⟶& 0\mathrm{as}(t_2-t_1)\to 0\mathrm{independently}\mathrm{of}(x,y)\in \mathbb{O},\hfill \end{array}$$

where ${\mathbb{J}}_i,i=1,2,3,4,$ are given in (11). Analogously, it can be shown that

$$\begin{array}{c}\hfill |{\mathbb{A}}_2(x,y)\left(t_2\right)-{\mathbb{A}}_2(x,y)\left(t_1\right)|⟶0\mathrm{as}(t_2-t_1)\to 0\mathrm{independently}\mathrm{of}(x,y)\in \mathbb{O}.\end{array}$$

Thus, ${\mathbb{A}}_1\left(\mathbb{O}\right)$ and ${\mathbb{A}}_2\left(\mathbb{O}\right)$ are equicontinuous, and hence $\mathbb{A}\left(\mathbb{O}\right)$ is equicontinuous. Hence, we deduce using the Arzelá–Ascoli theorem that $\mathbb{A}\left(\mathbb{O}\right)$ is completely continuous.

Finally, we consider a set $\mathbb{H}=\left\{\right(x,y)\in \mathbb{X}\times \mathbb{X}:(x,y)=\tau \mathbb{A}(x,y),0<\tau \le 1\}$ and verify its boundedness. Let $(x,y)\in \mathbb{H}.$ Then, $(x,y)=\tau \mathbb{A}(x,y)$ implies that $x\left(t\right)=\tau {\mathbb{A}}_1(x,y)\left(t\right)$ and $y\left(t\right)=\tau {\mathbb{A}}_2(x,y)\left(t\right)$ for $t\in \mathcal{J}.$ Then, according to assumption $\left(H_1\right)$, we have

$$\begin{array}{ccc}\hfill \parallel x\parallel & =& \underset{t\in \mathcal{J}}{sup}\left|x\left(t\right)\right|\le \underset{t\in \mathcal{J}}{sup}\left|{\mathbb{A}}_1(x,y)\right(t\left)\right|\hfill \\ & \le & \underset{t\in \mathcal{J}}{sup}\{{\int }_1^t\left[\frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left({\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\right)}{s}+{\chi }_1\frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +|{\mathbb{M}}_1\left(t\right)\left|\right\{\sum _{i=1}^{m_1}\left|{\lambda }_i\right|{\int }_1^{{\eta }_i}\left[\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}\frac{\left({\widehat{n}}_0+{\widehat{n}}_1\left|x\right|+{\widehat{n}}_2\left|y\right|\right)}{s}+{\chi }_2\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}\frac{\left|y\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_1}\left[\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left({\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\right)}{s}+{\chi }_1\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_2\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\rho }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left({\widehat{n}}_0+{\widehat{n}}_1\left|x\right|+{\widehat{n}}_2\left|y\right|\right)}{u}+{\chi }_2\frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(u\left)\right|}{u}\right]duds\hfill \\ \hfill & & +{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left({\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\right)}{s}+{\chi }_1\frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_3\left(t\right)\left|\right\{\sum _{l=1}^{m_2}\left|{\sigma }_l\right|{\int }_1^{{\nu }_l}\left[\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}\frac{\left({\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\right)}{s}+{\chi }_1\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}\frac{\left|x\right(s\left)\right|}{s}\right]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_2}\left[\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left({\widehat{n}}_0+{\widehat{n}}_1\left|x\right|+{\widehat{n}}_2\left|y\right|\right)}{s}+{\chi }_2\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(s\left)\right|}{s}\right]ds\}\hfill \\ \hfill & & +|{\mathbb{M}}_4\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\tau }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left({\widehat{m}}_0+{\widehat{m}}_1\left|x\right|+{\widehat{m}}_2\left|y\right|\right)}{u}+{\chi }_1\frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\left|x\right(u\left)\right|}{u}\right]duds\hfill \\ \hfill & & +{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left({\widehat{n}}_0+{\widehat{n}}_1\left|x\right|+{\widehat{n}}_2\left|y\right|\right)}{s}+{\chi }_2\frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\left|y\right(s)|}{s}\right]ds\left\}\right\},\hfill \end{array}$$

which implies that

$$\parallel x\parallel \le ({\widehat{m}}_0{\Psi }_1+{\widehat{n}}_0{\Psi }_2)+({\widehat{m}}_1{\Psi }_1+{\widehat{n}}_1{\Psi }_2+{\Psi }_3)\parallel x\parallel +({\widehat{m}}_2{\Psi }_1+{\widehat{n}}_2{\Psi }_2+{\Psi }_4)\parallel y\parallel .$$

(20)

In a similar manner, we can find that

$$\parallel y\parallel \le ({\widehat{m}}_0{\Phi }_1+{\widehat{n}}_0{\Phi }_2)+({\widehat{m}}_1{\Phi }_1+{\widehat{n}}_1{\Phi }_2+{\Phi }_3)\parallel x\parallel +({\widehat{m}}_2{\Phi }_1+{\widehat{n}}_2{\Phi }_2+{\Phi }_4)\parallel y\parallel .$$

(21)

From (20) and (21), it follows that

$$\parallel x\parallel +\parallel y\parallel \le {\widehat{m}}_0({\Psi }_1+{\Phi }_1)+{\widehat{n}}_0({\Psi }_2+{\Phi }_2)+max({\mathbb{S}}_1,{\mathbb{S}}_2)\parallel (x,y)\parallel ,$$

where ${\mathbb{S}}_1$ and ${\mathbb{S}}_2$ are given in (17). The above inequality can alternatively be written as

$$\parallel (x,y)\parallel \le {\displaystyle \frac{1}{{\mathbb{G}}_0}}[{\widehat{m}}_0({\Psi }_1+{\Phi }_1)+{\widehat{n}}_0({\Psi }_2+{\Phi }_2)],$$

where

$${\mathbb{G}}_0=1-max({\mathbb{S}}_1,{\mathbb{S}}_2).$$

Thus, the set $\mathbb{H}$ is bounded. Consequently, the operator $\mathbb{A}$ has at least one fixed point through application of the Leray–Schauder alternative [40]. Therefore, problem (1) and (2) admit at least one solution on $\mathcal{J}.$ □

Next, we obtain a result on the uniqueness of the problem (1) and (2) with the aid of Banach’s fixed point theorem [40].

Theorem 3.

Assume that $f_1,f_2\in C(\mathcal{J}\times \mathbb{R}\times \mathbb{R},\mathbb{R}).$ In addition, we suppose that the condition $\left(H_2\right)$ is satisfied and

$${\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)+max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)<1,$$

(22)

where ${\Psi }_p$ and ${\Phi }_p,$ $p=1,2,3,4,$ are given in (16). Then, the problem (1) and (2) have a unique solution on $\mathcal{J}.$

Proof. 

To verify the hypotheses of Banach’s fixed point theorem, we consider a closed ball ${\mathbb{B}}_r=\{(x,y)\in \mathbb{X}\times \mathbb{X}:\parallel (x,y)\parallel \le r\}$ with

$$r\ge {\displaystyle \frac{{\mathcal{M}}_1({\Psi }_1+{\Phi }_1)+{\mathcal{M}}_2({\Psi }_2+{\Phi }_2)}{1-\left[{\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)\right]-max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)}},$$

(23)

where ${sup}_{t\in \mathcal{J}}\left|f_1(t,0,0)\right|={\mathcal{M}}_1<\infty$ and ${sup}_{t\in \mathcal{J}}\left|f_2(t,0,0)\right|={\mathcal{M}}_2<\infty .$ Now, we establish that $\mathbb{A}{\mathbb{B}}_r\subset {\mathbb{B}}_r$, where $\mathbb{A}:{\mathbb{B}}_r⟶\mathbb{X}\times \mathbb{X}$ is given by (13). According to $\left(H_2\right)$, we have

$$\begin{array}{ccc}\hfill |f_1(t,x\left(t\right),y\left(t\right))|& =& |f_1(t,x\left(t\right),y\left(t\right))-f_1(t,0,0)+f_1(t,0,0)|\le {\mathbb{L}}_1(\parallel x\parallel +\parallel y\parallel )+{\mathcal{M}}_1\hfill \\ & \le & {\mathbb{L}}_{1}r+{\mathcal{M}}_1,\hfill \\ \hfill |f_2(t,x\left(t\right),y\left(t\right)|& =& |f_2(t,x\left(t\right),y\left(t\right))-f_2(t,0,0)+f_2(t,0,0)|\le {\mathbb{L}}_2(\parallel x\parallel +\parallel y\parallel )+{\mathcal{M}}_2\hfill \\ & \le & {\mathbb{L}}_{2}r+{\mathcal{M}}_2.\hfill \end{array}$$

(24)

For $(x,y)\in {\mathbb{B}}_r,$ it follows by using (24) that

$$\begin{array}{ccc}\hfill \parallel {\mathbb{A}}_1(x,y)\parallel & \le & \underset{t\in \mathcal{J}}{sup}\{{\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right)}{s}}+{\chi }_1{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\left|x\right(s\left)\right|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_1\left(t\right)\left|\right\{\sum _{i=1}^{m_1}\left|{\lambda }_i\right|{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right)}{s}}+{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{\left|y\right(s\left)\right|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right)}{s}}+{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\left|x\right(s\left)\right|}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_2\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\rho }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right)}{u}}+{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\left|y\right(u\left)\right|}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right)}{s}}+{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\left|x\right(s\left)\right|}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_3\left(t\right)\left|\right\{\sum _{l=1}^{m_2}\left|{\sigma }_l\right|{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right)}{s}}+{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{\left|x\right(s\left)\right|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right)}{s}}+{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\left|y\right(s\left)\right|}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_4\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\tau }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right)}{u}}+{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\left|x\right(u\left)\right|}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right)}{s}}+{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\left|y\right(s\left|\right)}{s}}\right]ds\left\}\right\}\hfill \\ & & \le \left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right){\Psi }_1+{\Psi }_3\parallel x\parallel +\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right){\Psi }_2+{\Psi }_4\parallel y\parallel .\hfill \end{array}$$

(25)

Likewise, we can find that

$$\parallel {\mathbb{A}}_2(x,y)\parallel \le \left({\mathbb{L}}_{1}r+{\mathcal{M}}_1\right){\Phi }_1+{\Phi }_3\parallel x\parallel +\left({\mathbb{L}}_{2}r+{\mathcal{M}}_2\right){\Phi }_2+{\Phi }_4\parallel y\parallel .$$

(26)

From (25) and (26), together with (23),

$$\begin{array}{ccc}\hfill \parallel \mathbb{A}(x,y)\parallel & =& \parallel {\mathbb{A}}_1(x,y)\parallel +\parallel {\mathbb{A}}_2(x,y)\parallel \hfill \\ & & \le \left[{\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)\right]r+{\mathcal{M}}_1({\Psi }_1+{\Phi }_1)+{\mathcal{M}}_2({\Psi }_2+{\Phi }_2)\hfill \\ & & +max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)r\le r,\hfill \end{array}$$

which shows that $\mathbb{A}(x,y)\in {\mathbb{B}}_r$. Hence, $\mathbb{A}{\mathbb{B}}_r\subset {\mathbb{B}}_r$ since $(x,y)\in {\mathbb{B}}_r$ is an arbitrary element.

Next, we verify that the operator $\mathbb{A}$ is a contraction. For this, let $(x_1,y_1),(x_2,y_2)\in \mathbb{X}\times \mathbb{X}.$ Then, for any $t\in \mathcal{J}$, we obtain

$$\begin{array}{ccc}\hfill & & \parallel {\mathbb{A}}_1(x_2,y_2)-{\mathbb{A}}_1(x_1,y_1)\parallel \hfill \\ & \le & \underset{t\in \mathcal{J}}{sup}\left\{{\int }_1^t\right[\frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x_2\left(s\right),y_2\left(s\right))-f_1(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_1\frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{|x_2\left(s\right)-x_1\left(s\right)|}{s}]ds\hfill \\ \hfill & & +|{\mathbb{M}}_1\left(t\right)\left|\right\{\sum _{i=1}^{m_1}\left|{\lambda }_i\right|{\int }_1^{{\eta }_i}[\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x_2\left(s\right),y_2\left(s\right))-f_2(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_2\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}\frac{|y_2\left(s\right)-y_1\left(s\right)|}{s}]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_1}[\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x_2\left(s\right),y_2\left(s\right))-f_1(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_1\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{|x_2\left(s\right)-x_1\left(s\right)|}{s}\left]ds\right\}\hfill \\ \hfill & & +|{\mathbb{M}}_2\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\rho }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s[\frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(u,x_2\left(u\right),y_2\left(u\right))-f_2(u,x_1\left(u\right),y_1\left(u\right))|}{u}\hfill \\ & & +{\chi }_2\frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{|y_2\left(u\right)-y_1\left(u\right)|}{u}]duds\hfill \\ \hfill & & +{\int }_1^T[\frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x_2\left(s\right),y_2\left(s\right))-f_1(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_1\frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{|x_2\left(s\right)-x_1\left(s\right)|}{s}\left]ds\right\}\hfill \\ \hfill & & +|{\mathbb{M}}_3\left(t\right)\left|\right\{\sum _{l=1}^{m_2}\left|{\sigma }_l\right|{\int }_1^{{\nu }_l}[\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}\frac{|f_1(s,x_2\left(s\right),y_2\left(s\right))-f_1(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_1\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}\frac{|x_2\left(s\right)-x_1\left(s\right)|}{s}]ds\hfill \\ \hfill & & +{\int }_1^{{\mu }_2}[\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x_2\left(s\right),y_2\left(s\right))-f_2(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_2\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{|y_2\left(s\right)-y_1\left(s\right)|}{s}\left]ds\right\}\hfill \\ \hfill & & +|{\mathbb{M}}_4\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\tau }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s[\frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{|f_1(u,x_2\left(u\right),y_2\left(u\right))-f_1(u,x_1\left(u\right),y_1\left(u\right))|}{u}\hfill \\ & & +{\chi }_1\frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{|x_2\left(u\right)-x_1\left(u\right)|}{u}]duds\hfill \\ \hfill & & +{\int }_1^T[\frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{|f_2(s,x_2\left(s\right),y_2\left(s\right))-f_2(s,x_1\left(s\right),y_1\left(s\right))|}{s}\hfill \\ & & +{\chi }_2\frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{|y_2\left(s\right)-y_1\left(s\right)|}{s}\left]ds\right\}\}\hfill \\ \hfill & & \\ & \le & \left({\mathbb{L}}_1{\Psi }_1+{\mathbb{L}}_2{\Psi }_2\right)(\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel )+{\Psi }_3\parallel x_2-x_1\parallel +{\Psi }_4\parallel y_2-y_1\parallel .\hfill \end{array}$$

(27)

Similarly, we can find that

$$\parallel {\mathbb{A}}_2(x_2,y_2)-{\mathbb{A}}_2(x_1,y_1)\parallel \le \left({\mathbb{L}}_1{\Phi }_1+{\mathbb{L}}_2{\Phi }_2\right)(\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel )+{\Phi }_3\parallel x_2-x_1\parallel +{\Phi }_4\parallel y_2-y_1\parallel .$$

(28)

From (27) and (28), we have

$$\begin{array}{ccc}& & \parallel \mathbb{A}(x_2,y_2)-\mathbb{A}(x_1,y_1)\parallel \hfill \\ & \le & \left[{\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)+max({\Psi }_3+{\Phi }_3,Z_4+N_4)\right](\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel ),\hfill \end{array}$$

which, according to (22), implies that $\mathbb{A}$ is a contraction. Therefore, a unique fixed point exists for the operator $\mathbb{A}$ through the application of Banach’s fixed point theorem. Thus, a unique solution to problem (1) and (2) exists on $\mathcal{J}.$ □

4. Examples

Here, we present examples illustrating the results obtained in the previous section.

Example 1.

Consider a coupled system of nonlinear Langevin equations

$$\left\{\begin{array}{c}{\displaystyle {}^{HH}D^{{\kappa }_1,{\xi }_1}({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)x\left(t\right)=f_1(t,x\left(t\right),y\left(t\right)),t\in \mathcal{J},}\hfill \\ {\displaystyle {}^{HH}D^{{\kappa }_3,{\xi }_3}({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)y\left(t\right)=f_2(t,x\left(t\right),y\left(t\right)),t\in \mathcal{J},}\hfill \end{array}\right.$$

(29)

subject to the boundary data

$$\left\{\begin{array}{c}{\displaystyle x\left(1\right)=0,x\left({\mu }_1\right)=\sum _{i=1}^3{\lambda }_i{}^{H}I^{{\alpha }_i}y\left({\eta }_i\right),x\left(T\right)=\sum _{j=1}^3{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}y\left(s\right)ds,}\hfill \\ {\displaystyle y\left(1\right)=0,y\left({\mu }_2\right)=\sum _{l=1}^2{\sigma }_l{}^{H}I^{{\beta }_l}x\left({\nu }_l\right),y\left(T\right)=\sum _{j=1}^3{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}x\left(s\right)ds,}\hfill \end{array}\right.$$

(30)

with ${\kappa }_1=1.5$, ${\kappa }_2=0.5$, ${\kappa }_3=1.25$, ${\kappa }_4=0.6$, $0\le {\xi }_p\le 1(p=1,2,3,4)$, ${\gamma }_1=1.75$, ${\gamma }_2=1.5$, ${\gamma }_3=0.75$, ${\gamma }_4=0.7$, ${\alpha }_1=5/3$, ${\alpha }_2=1/3$, ${\alpha }_3=8/3$, ${\beta }_1=8/5$, ${\beta }_2=2/5$, $v_1=4/3$, $v_2=5/3$, ${\zeta }_1=6/3$, ${\zeta }_2=8/3$, ${\zeta }_3=10/3$, ${\chi }_1=1/20$, ${\chi }_2=1/50$, ${\omega }_1=7/3$, ${\omega }_2=9/3$, ${\omega }_3=11/3$, ${\eta }_1=12/3$, ${\eta }_2=13/3$, ${\eta }_3=14/3$, ${\sigma }_1=1/15$, ${\sigma }_2=1/10$, ${\lambda }_1=1/20$, ${\lambda }_2=1/4$, ${\lambda }_3=1/4$, ${\rho }_1=1/12$, ${\rho }_2=1/6$, ${\rho }_3=1/4$, ${\tau }_1=1/7$, ${\tau }_2=1/5$, ${\tau }_3=1/2$, ${\mu }_1=1.1$, ${\mu }_2=1.2$, $n=3$, $m_1=3$, $m_2=2,$, $\mathcal{J}=[1,5]$.

Using the given values, we find that ${\Psi }_1\approx 2.74585874$, ${\Psi }_2\approx 2.84924444$, ${\Psi }_3\approx 0.23579603$, ${\Psi }_4\approx 0.09318252$, ${\Phi }_1\approx 0.13336712$, ${\Phi }_2\approx 3.17256265$, ${\Phi }_3\approx 0.02135806$, and ${\Phi }_4\approx 0.09118365$ (${\Psi }_p$ and ${\Phi }_p$, $p=1,2,3,4$, are given in (16)).

(a)

To illustrate Theorem 2, we take

$$\begin{array}{ccc}\hfill & & {\displaystyle f_1(t,x\left(t\right),y\left(t\right))=e^{-5t}+\frac{cos x\left(t\right)}{20t+4}+\frac{\left|y\right(t\left)\right|}{(t+20)(1+|y\left(t\right)\left|\right)},t\in \mathcal{J},}\hfill \end{array}$$

$$\begin{array}{ccc}& & {\displaystyle f_2(t,x\left(t\right),y\left(t\right))=\frac{1}{\sqrt{3t+1}}+\frac{x\left(t\right)siny\left(t\right)}{10t+20}+\frac{\left|y\left(t\right)\right|}{{(t+4)}^3\left(\sqrt{1+{\left|x\left(t\right)\right|}^2}\right)},t\in \mathcal{J},}\hfill \end{array}$$

(31)

and note that

$$|f_1(t,x\left(t\right),y\left(t\right))|\le {\displaystyle \frac{1}{e^5}}+{\displaystyle \frac{1}{24}}\left|x\left(t\right)\right|+{\displaystyle \frac{1}{21}}\left|y\left(t\right)\right|,$$

$$|f_2(t,x\left(t\right),y\left(t\right))|\le {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{50}}\left|x\left(t\right)\right|+{\displaystyle \frac{1}{125}}\left|y\left(t\right)\right|,$$

that is, the assumption $\left(H_1\right)$ holds with ${\widehat{m}}_0={\displaystyle \frac{1}{e^5}}$, ${\widehat{m}}_1={\displaystyle \frac{1}{24}}$, ${\widehat{m}}_2={\displaystyle \frac{1}{21}}$, ${\widehat{n}}_0={\displaystyle \frac{1}{2}}$, ${\widehat{n}}_1={\displaystyle \frac{1}{30}}$, ${\widehat{n}}_2={\displaystyle \frac{1}{125}}$.

Moreover, we find that ${\mathbb{S}}_1\approx 0.57784874<1$ and ${\mathbb{S}}_2\approx 0.36964662<1$. Thus, the hypotheses of Theorem 2 are verified. Therefore, the conclusion of Theorem 2 applies to the problem (29) and (30), with $f_1$ and $f_2$ given in (31).

(b)

To illustrate Theorem 3, we take

$$\begin{array}{c}\hfill f_1(t,x\left(t\right),y\left(t\right))={\displaystyle \frac{1}{5(t+8)}}{tan}^{-1}x\left(t\right)+{\displaystyle \frac{cosy\left(t\right)}{5\sqrt{(t+80)}}}+e^t,t\in \mathcal{J},\end{array}$$

$$\begin{array}{c}\hfill f_2(t,x\left(t\right),y\left(t\right))={\displaystyle \frac{1}{10\sqrt{2t+7}}}\left({\displaystyle \frac{\left|x\right(t\left)\right|}{1+\left|x\right(t\left)\right|}}+siny\left(t\right)\right)+t^2,t\in \mathcal{J}.\end{array}$$

(32)

For each $t\in \mathcal{J}$ and $x_1,y_1,x_2,y_2\in \mathbb{R}$, we notice that

$$|f(t,x_2,y_2)-f(t,x_1,y_1)|\le {\displaystyle \frac{1}{45}}\left(\right|x_2-x_1|+|y_2-y_1\left|\right),$$

$$|g(t,x_2,y_2)-g(t,x_1,y_1)|\le {\displaystyle \frac{1}{30}}\left(\right|x_2-x_1|+|y_2-y_1\left|\right).$$

Thus, the condition ($H_2$) holds true with ${\mathbb{L}}_1={\displaystyle \frac{1}{45}}$ and ${\mathbb{L}}_2={\displaystyle \frac{1}{30}}$. Further, we have

$$\begin{array}{c}\hfill {\displaystyle {\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)+max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)\approx 0.52186379<1.}\end{array}$$

Clearly, the hypothesis of Theorem 3 is satisfied. Hence, the problem (29) and (30) with the nonlinear functions $f_1$ and $f_2$ given by (32) have a unique solution on $\mathcal{J}$.

5. The Stability Analysis

Let us first develop the arguments for the Ulam–Hyers stability [41] of the problem (1) and (2).

For an arbitrary $ϵ=({ϵ}_1,{ϵ}_2)>0$ and $t\in \mathcal{J},$ we consider a system of inequalities with the boundary conditions (2) given by

$$\left\{\begin{array}{c}{\displaystyle |{}^{HH}D^{{\kappa }_1,{\xi }_1}({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)\widehat{x}\left(t\right)-f_1(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))|\le {ϵ}_1,}\hfill \\ {\displaystyle |{}^{HH}D^{{\kappa }_3,{\xi }_3}({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)\widehat{y}\left(t\right)-f_2(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))|\le {ϵ}_2.}\hfill \end{array}\right.$$

(33)

If $(\widehat{x},\widehat{y})\in \mathbb{X}\times \mathbb{X}$ is a solution of the system of inequalities (33) with the boundary conditions (2), then functions ${\lambda }_1,{\lambda }_2\in C(\mathcal{J},\mathbb{R})$ exist such that $\left|{\lambda }_1\left(t\right)\right|\le {ϵ}_1$, $|{\lambda }_2\left(t\right)|\le {ϵ}_2$, $t\in \mathcal{J}$ and such that $(\widehat{x},\widehat{y})\in \mathbb{X}\times \mathbb{X}$ satisfies the system of Hilfer–Hadamard-type fractional Langevin equations

$$\left\{\begin{array}{c}{\displaystyle {}^{HH}D^{{\kappa }_1,{\xi }_1}({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)\widehat{x}\left(t\right)=f_1(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))+{\lambda }_1\left(t\right),}\hfill \\ {\displaystyle {}^{HH}D^{{\kappa }_3,{\xi }_3}({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)\widehat{y}\left(t\right)=f_2(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))+{\lambda }_2\left(t\right).}\hfill \end{array}\right.$$

Thus, for the forthcoming analysis, we consider the boundary value problem

$$\left\{\begin{array}{c}{\displaystyle {}^{HH}D^{{\kappa }_1,{\xi }_1}({}^{HH}D^{{\kappa }_2,{\xi }_2}+{\chi }_1)\widehat{x}\left(t\right)=f_1(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))+{\lambda }_1\left(t\right),t\in \mathcal{J},}\hfill \\ {\displaystyle {}^{HH}D^{{\kappa }_3,{\xi }_3}({}^{HH}D^{{\kappa }_4,{\xi }_4}+{\chi }_2)\widehat{y}\left(t\right)=f_2(t,\widehat{x}\left(t\right),\widehat{y}\left(t\right))+{\lambda }_2\left(t\right),t\in \mathcal{J},}\hfill \\ {\displaystyle \widehat{x}\left(1\right)=0,\widehat{x}\left({\mu }_1\right)=\sum _{i=1}^{m_1}{\lambda }_i{}^{H}I^{{\alpha }_i}\widehat{y}\left({\eta }_i\right),\widehat{x}\left(T\right)=\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}\widehat{y}\left(s\right)ds,}\hfill \\ {\displaystyle \widehat{y}\left(1\right)=0,\widehat{y}\left({\mu }_2\right)=\sum _{l=1}^{m_2}{\sigma }_l{}^{H}I^{{\beta }_l}\widehat{x}\left({\nu }_l\right),\widehat{y}\left(T\right)=\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}\widehat{x}\left(s\right)ds.}\hfill \end{array}\right.$$

(34)

Definition 4.

The system (1) and (2) is called Ulam–Hyers-stable if we can find $c=(c_{f_1},c_{f_2})>0$ such that for each solution $(\widehat{x},\widehat{y})\in \mathbb{X}\times \mathbb{X}$ of (34), a unique solution $(x,y)\in \mathbb{X}\times \mathbb{X}$ to the system (1) and (2) exists satisfying

$$\parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le c{ϵ}^T,t\in \mathcal{J}.$$

Definition 5.

The system (1) and (2) is said to be generalized Ulam–Hyers-stable if a unique solution $(x,y)\in \mathbb{X}\times \mathbb{X}$ to the system (1) and (2) exists satisfying

$$\parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le \Omega \left(ϵ\right),t\in \mathcal{J},$$

for each solution $(\widehat{x},\widehat{y})\in \mathbb{X}\times \mathbb{X}$ of (34), where $\Omega =({\Omega }_1,{\Omega }_2)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ with $\Omega \left(0\right)=0\left({\Omega }_1\left(0\right)={\Omega }_2\left(0\right)=0\right)$.

Theorem 4.

Let $f_1,f_2\in C(\mathcal{J}\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ and the assumption $\left(H_2\right)$ and the condition (22) hold. Then, the system (1) and (2) is Ulam–Hyers-stable and hence generalized Ulam–Hyers-stable in $\mathbb{X}\times \mathbb{X}$.

Proof. 

According to Lemma 2, we can write the solution to (34) as

$$\begin{array}{ccc}\hfill \widehat{x}\left(t\right)& =& {\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\hfill \\ & & +{\mathbb{M}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left(f_2(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))+{\lambda }_2\left(u\right)\right)}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & +{\mathbb{M}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{\left(f_1(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))+{\lambda }_1\left(u\right)\right)}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \widehat{y}\left(t\right)& =& {\int }_1^t\left[\frac{{(log\frac{t}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}-{\chi }_2\frac{{(log\frac{t}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\widehat{y}\left(s\right)}{s}\right]ds\hfill \\ & & +{\mathbb{N}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}\frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}-{\chi }_2\frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}\frac{\widehat{y}\left(s\right)}{s}\right]ds\hfill \\ \hfill & & -{\int }_1^{{\mu }_1}\left[\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}-{\chi }_1\frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\widehat{x}\left(s\right)}{s}\right]ds\}\hfill \\ \hfill & & +{\mathbb{N}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left(f_2(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))+{\lambda }_2\left(u\right)\right)}{u}-{\chi }_2\frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\widehat{y}\left(u\right)}{u}\right]duds\hfill \\ \hfill & & -{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}-{\chi }_1\frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\widehat{x}\left(s\right)}{s}\right]ds\}\hfill \\ \hfill & & +{\mathbb{N}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}\frac{\left(f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_1\left(s\right)\right)}{s}-{\chi }_1\frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}\frac{\widehat{x}\left(s\right)}{s}\right]ds\hfill \\ \hfill & & -{\int }_1^{{\mu }_2}\left[\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}-{\chi }_2\frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\widehat{y}\left(s\right)}{s}\right]ds\}\hfill \\ \hfill & & +{\mathbb{N}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}{\int }_1^s\left[\frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}\frac{\left(f_1(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))+{\lambda }_1\left(u\right)\right)}{u}-{\chi }_1\frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}\frac{\widehat{x}\left(u\right)}{u}\right]duds\hfill \\ \hfill & & -{\int }_1^T\left[\frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}\frac{\left(f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))+{\lambda }_2\left(s\right)\right)}{s}-{\chi }_2\frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}\frac{\widehat{y}\left(s\right)}{s}\right]ds\}.\hfill \end{array}$$

In view of $|{\lambda }_1|<{ϵ}_1$, $|{\lambda }_2|<{ϵ}_2$ and (16), we have

$$\begin{array}{ccc}& & \underset{t\in \mathcal{J}}{sup}|\widehat{x}\left(t\right)-{\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\hfill \\ & & -{\mathbb{M}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{M}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{M}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{M}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\left\}\right|\le {\Psi }_1{ϵ}_1+{\Psi }_2{ϵ}_2,\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \underset{t\in \mathcal{J}}{sup}|\widehat{y}\left(t\right)-{\int }_1^t\left[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\hfill \\ & & -{\mathbb{N}}_1\left(t\right)\{\sum _{i=1}^{m_1}{\lambda }_i{\int }_1^{{\eta }_i}\left[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_1}\left[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{N}}_2\left(t\right)\{\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))}{u}}-{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{N}}_3\left(t\right)\{\sum _{l=1}^{m_2}{\sigma }_l{\int }_1^{{\nu }_l}\left[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{\widehat{x}\left(s\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^{{\mu }_2}\left[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\}\hfill \\ \hfill {\displaystyle }& & -{\mathbb{N}}_4\left(t\right)\{\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s\left[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{f_1(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))}{u}}-{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{\widehat{x}\left(u\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & -{\int }_1^T\left[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))}{s}}-{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{\widehat{y}\left(s\right)}{s}}\right]ds\left\}\right|\le {\Phi }_1{ϵ}_1+{\Phi }_2{ϵ}_2.\hfill \end{array}$$

According to $\left(H_2\right)$, we have

$$\begin{array}{ccc}\hfill \parallel \widehat{x}-x\parallel & =& \underset{t\in \mathcal{J}}{sup}|\widehat{x}\left(t\right)-x\left(t\right)|\le {\Psi }_1{ϵ}_1+{\Psi }_2{ϵ}_2\hfill \\ & & +\underset{t\in \mathcal{J}}{sup}\left\{{\int }_1^t\right[{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{|f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_1(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_1{\displaystyle \frac{{(log\frac{t}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{|\widehat{x}\left(s\right)-x\left(s\right)|}{s}}]ds\hfill \\ & & +|{\mathbb{M}}_1\left(t\right)\left|\right\{\sum _{i=1}^{m_1}\left|{\lambda }_i\right|{\int }_1^{{\eta }_i}[{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{|f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_2(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_2{\displaystyle \frac{{(log\frac{{\eta }_i}{s})}^{{\alpha }_i+{\kappa }_4-1}}{\Gamma ({\alpha }_i+{\kappa }_4)}}{\displaystyle \frac{|\widehat{y}\left(s\right)-y\left(s\right)|}{s}}]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^{{\mu }_1}[{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{|f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_1(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_1{\displaystyle \frac{{(log\frac{{\mu }_1}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{|\widehat{x}\left(s\right)-x\left(s\right)|}{s}}\left]ds\right\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_2\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\rho }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{|f_2(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))-f_2(u,x\left(u\right),y\left(u\right))|}{u}}\hfill \\ & & +{\chi }_2{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{|\widehat{y}\left(u\right)-y\left(u\right)|}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^T[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{|f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_1(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_1{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{|\widehat{x}\left(s\right)-x\left(s\right)|}{s}}\left]ds\right\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_3\left(t\right)\left|\right\{\sum _{l=1}^{m_2}\left|{\sigma }_l\right|{\int }_1^{{\nu }_l}[{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{|f_1(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_1(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_1{\displaystyle \frac{{(log\frac{{\nu }_l}{s})}^{{\beta }_l+{\kappa }_2-1}}{\Gamma ({\beta }_l+{\kappa }_2)}}{\displaystyle \frac{|\widehat{x}\left(s\right)-x\left(s\right)|}{s}}]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^{{\mu }_2}[{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{|f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_2(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_2{\displaystyle \frac{{(log\frac{{\mu }_2}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{|\widehat{y}\left(s\right)-y\left(s\right)|}{s}}\left]ds\right\}\hfill \\ \hfill {\displaystyle }& & +|{\mathbb{M}}_4\left(t\right)\left|\right\{\sum _{j=1}^n\left|{\tau }_j\right|{\int }_{{\zeta }_j}^{{\omega }_j}{\displaystyle \frac{1}{s}}{\int }_1^s[{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_1+{\kappa }_2-1}}{\Gamma ({\kappa }_1+{\kappa }_2)}}{\displaystyle \frac{|f_1(u,\widehat{x}\left(u\right),\widehat{y}\left(u\right))-f_1(u,x\left(u\right),y\left(u\right))|}{u}}\hfill \\ & & +{\chi }_1{\displaystyle \frac{{(log\frac{s}{u})}^{{\kappa }_2-1}}{\Gamma \left({\kappa }_2\right)}}{\displaystyle \frac{|\widehat{x}\left(u\right)-x\left(u\right)|}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +{\int }_1^T[{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_3+{\kappa }_4-1}}{\Gamma ({\kappa }_3+{\kappa }_4)}}{\displaystyle \frac{|f_2(s,\widehat{x}\left(s\right),\widehat{y}\left(s\right))-f_2(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ & & +{\chi }_2{\displaystyle \frac{{(log\frac{T}{s})}^{{\kappa }_4-1}}{\Gamma \left({\kappa }_4\right)}}{\displaystyle \frac{|\widehat{y}\left(s\right)-y\left(s\right)|}{s}}\left]ds\right\}\}\hfill \\ & \le & {\Psi }_1{ϵ}_1+{\Psi }_2{ϵ}_2+\left({\mathbb{L}}_1{\Psi }_1+{\mathbb{L}}_2{\Psi }_2\right)(\parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel )+{\Psi }_3\parallel \widehat{x}-x\parallel +{\Psi }_4\parallel \widehat{y}-y\parallel ,\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel \widehat{x}-x\parallel \le {\Psi }_1{ϵ}_1+{\Psi }_2{ϵ}_2+\left({\mathbb{L}}_1{\Psi }_1+{\mathbb{L}}_2{\Psi }_2\right)(\parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel )+{\Psi }_3\parallel \widehat{x}-x\parallel +{\Psi }_4\parallel \widehat{y}-y\parallel .\end{array}$$

In a similar fashion, one can find that

$$\begin{array}{c}\hfill \parallel \widehat{y}-y\parallel \le {\Phi }_1{ϵ}_1+{\Phi }_2{ϵ}_2+\left({\mathbb{L}}_1{\Phi }_1+{\mathbb{L}}_2{\Phi }_2\right)(\parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel )+{\Phi }_3\parallel \widehat{x}-x\parallel +{\Phi }_4\parallel \widehat{y}-y\parallel .\end{array}$$

Thus, according to condition (22), we obtain

$$\begin{array}{ccc}& & \parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le \parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel \hfill \\ & & \le ({\Psi }_1+{\Phi }_1){ϵ}_1+({\Psi }_2+{\Phi }_2){ϵ}_2+\left({\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)\right)\left(\parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel \right)\hfill \\ & & +max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)(\parallel \widehat{x}-x\parallel +\parallel \widehat{y}-y\parallel ).\hfill \end{array}$$

Therefore,

$$\parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le {\displaystyle \frac{({\Psi }_1+{\Phi }_1){ϵ}_1+({\Psi }_2+{\Phi }_2){ϵ}_2}{1-\left[{\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)\right]-max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)}}.$$

(35)

Letting ${\displaystyle c=(c_{f_1},c_{f_2})=\left(\frac{{\Psi }_1+{\Phi }_1}{1-\mathbb{T}},\frac{{\Psi }_2+{\Phi }_2}{1-\mathbb{T}}\right)}$, we obtain

$$\parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le c{ϵ}^T,$$

where $\mathbb{T}=\left[{\mathbb{L}}_1({\Psi }_1+{\Phi }_1)+{\mathbb{L}}_2({\Psi }_2+{\Phi }_2)\right]+max({\Psi }_3+{\Phi }_3,{\Psi }_4+{\Phi }_4)<1,$ according to condition (22). Hence, the system (1) and (2) is Ulam–Hyers-stable. Furthermore, it is generalized Ulam–Hyers-stable since $\parallel (\widehat{x},\widehat{y})-(x,y)\parallel \le \Omega \left(ϵ\right)$ with $\Omega \left(ϵ\right)=c{ϵ}^T,\Omega \left(0\right)=0$. □

Example 2.

Problem (29) and (30) with $f_1$ and $f_2$ given in (32) is Ulam–Hyers-stable, as well as generalized Ulam–Hyers-stable, since $\mathbb{T}\approx 0.52186379<1$.

6. Conclusions

We have presented the criteria ensuring the existence and Ulam–Hyers stability of solutions to a fully coupled nonlocal integral boundary value problem of nonlinear fractional Hilfer–Hadamard Langevin equations. The results derived in this paper are not only novel in the given configuration but also correspond to some new configurations as special cases. For example, letting ${\rho }_j={\tau }_j=0$ for $j=1,\dots ,n$, the present results become those for boundary conditions

$$\left\{\begin{array}{c}{\displaystyle x\left(1\right)=0,x\left({\mu }_1\right)=\sum _{i=1}^{m_1}{\lambda }_i{}^{H}I_{1^{+}}^{{\alpha }_i}y\left({\eta }_i\right),x\left(T\right)=0,}\hfill \\ {\displaystyle y\left(1\right)=0,y\left({\mu }_2\right)=\sum _{l=1}^{m_2}{\sigma }_l{}^{H}I_{1^{+}}^{{\beta }_l}x\left({\nu }_l\right),y\left(T\right)=0.}\hfill \end{array}\right.$$

If we take ${\lambda }_i=0$ for $i=1,\dots ,m_1$ and ${\sigma }_l=0$ for $l=1,\dots ,m_2$ in the present results, we obtain results for the given system with the boundary conditions

$$\left\{\begin{array}{c}{\displaystyle x\left(1\right)=0,x\left({\mu }_1\right)=0,x\left(T\right)=\sum _{j=1}^n{\rho }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}y\left(s\right)ds,}\hfill \\ {\displaystyle y\left(1\right)=0,y\left({\mu }_2\right)=0,y\left(T\right)=\sum _{j=1}^n{\tau }_j{\int }_{{\zeta }_j}^{{\omega }_j}\frac{1}{s}x\left(s\right)ds.}\hfill \end{array}\right.$$

Our results correspond to the four-point nonlocal boundary conditions

$$x\left(1\right)=0,x\left({\mu }_1\right)=0,x\left(T\right)=0,y\left(1\right)=0,y\left({\mu }_2\right)=0,y\left(T\right)=0,$$

by taking all ${\lambda }_i=0(i=1,\dots ,m_1),{\sigma }_l=0(l=1,\dots ,m_2),{\rho }_j={\tau }_j=0(j=1,\dots ,n)$.