Study of a Coupled Integral–Multipoint Boundary Value Problem of Langevin–Type Nonlinear Riemann–Liouville and Hadamard Fractional Differential Equations

Abstract

Fractional Langevin models are found to be useful in the study of physical phenomena such as diffusion processes, gait variability, etc. Langevin equations involving different fractional–order operators and boundary conditions have been addressed by many researchers. In this article, we formulate a new Langevin model consisting of a coupled system of Riemann–Liouville and Hadamard–type nonlinear fractional differential equations and coupled multipoint–integral boundary conditions. We present the existence and Ulam–Hyers stability criteria for solutions of the given model problem. Our study is based on the tools of the fixed–point theory. Numerical examples with graphical representations of solutions are offered to demonstrate the application of the obtained results. Our work is novel and useful in the given configuration, and specializes to some new results.

1. Introduction

During the study of Brownian motion, the Newton’s second law of motion for stochastic physics presented by Langevin [1] is called the Langevin equation. However, the failure of this equation to deal with the complex systems describing the physical phenomena in anomalous transport [2], statistical mechanics [3], fluctuation–dissipation configuration [4], etc., led to its several generalizations.

The topic of fractional calculus has been extensively studied during the last two decades due to the application of its tools in diverse disciplines like neural networks [5], diffusion processes [6,7], synchronization [8], immune systems [9], ecological models [10], financial economics, etc. In contrast to the classical derivative, we do have many definitions of fractional derivatives like Riemann–Liouville, Caputo (Liouville–Caputo), Hadamard, Hilfer, etc. The Hadamard fractional derivative [11] contains a logarithmic function in its definition. For further details, see the text [12].

Let us now dwell on some applications of Riemann–Liouville and Hadamard fractional operators. The Riemann–Liouville fractional derivative operators are used to model complex dynamical systems depending on memory effects and nonlocal interactions like predator–prey systems [13], diffusive process [14], reaction–diffusion systems [15], etc. On the other hand, the Hadamard–type fractional integral and derivative operators, introduced in [16,17], find their applications in probability theory [18], generalization of the Lomnitz logarithmic creep law [19], nonlocal scaling dynamics [20], etc.

The fractional analogue of the Langevin equation was presented in [21] by replacing the ordinary derivative in it by its fractional counterpart, while the Langevin equation having two different fractional orders was outlined in [22]. Fractional Langevin models associated with diffusion process and gait variability are discussed in [23] and [24], respectively. Fractional-order Langevin equations equipped with a variety of boundary conditions have been addressed by many authors; for example, see [25,26,27]. For some recent work on boundary value problems for fractional Langevin equations, see [28,29,30]; one can find some results on Ulam–Hyers stability for fractional Langevin equations in [31,32]. The notion of Ulam–Hyers stability, introduced in [33] and refined in [34], appears in various fields, like functional equations [35], Black–Scholes equation [36], etc. In [37], the authors studied Hyers–Ulam–Rassias stability of fractional delay differential equations with delay.

Keeping in mind the importance of Riemann–Liouville and Hadamard fractional derivative operators described in the paragraph before the preceding one, a coupled system of Riemann–Liouville and Hadamard fractional Langevin equations with fractional integral boundary conditions was studied in [38]. In this paper, we enrich this study by considering a new class of coupled fractional Langevin equations of arbitrary orders containing Riemann–Liouville and Hadamard fractional derivative operators of the form

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D_{a^{+}}^{{\rho }_1}\left({}^{RL}D_{a^{+}}^{{\omega }_1}+{\mu }_1\left(\tau \right)\right)x\left(\tau \right)=f(\tau ,x\left(\tau \right),y\left(\tau \right)),{\rho }_1\in (0,1),{\omega }_1\in (n-1,n),n\ge 2,\tau \in \mathcal{T},}\hfill \\ {\displaystyle {}^{H}D_{a^{+}}^{{\rho }_2}\left({}^{H}D_{a^{+}}^{{\omega }_2}+{\mu }_2\left(\tau \right)\right)y\left(\tau \right)=g(\tau ,x\left(\tau \right),y\left(\tau \right)),{\rho }_2\in (0,1),{\omega }_2\in (m-1,m),m\ge 2,\tau \in \mathcal{T},}\hfill \end{array}\right.$$

(1)

supplemented with a new type of coupled integral–multipoint boundary conditions

$$\left\{\begin{array}{c}{\displaystyle x\left(a\right)=0,{}^{RL}D_{a^{+}}^{{\omega }_1-v_1}x\left(a\right)=0,v_1=1,2,3,\dots ,n-1,}\hfill \\ {\displaystyle y\left(a\right)=0,{}^{H}D_{a^{+}}^{{\omega }_2-v_2}y\left(a\right)=0,v_2=1,2,3,\dots ,m-1,}\hfill \\ {\displaystyle x\left(b\right)={\lambda }_1{\int }_a^b\frac{1}{s}y\left(s\right)ds+\sum _{i=1}^p{\eta }_{i}y\left({\alpha }_i\right),{\alpha }_i\in (a,b),{\lambda }_1\ge 0,{\eta }_i\in \mathbb{R},}\hfill \\ {\displaystyle y\left(b\right)={\lambda }_2{\int }_a^{b}x\left(s\right)ds+\sum _{j=1}^q{\zeta }_{j}x\left({\sigma }_j\right),{\sigma }_j\in (a,b),{\lambda }_2\ge 0,{\zeta }_j\in \mathbb{R}.}\hfill \end{array}\right.$$

(2)

where ${}^{RL}D_{a^{+}}^{(.)}$ and ${}^{H}D_{a^{+}}^{(.)}$, respectively, denote the Riemann–Liouville and Hadamard fractional derivative operators of order $(.)$, $\mathcal{T}=[a,b],a>0$, ${\mu }_1,{\mu }_2\in C(\mathcal{T},\mathbb{R})$ and $f,g\in C\left(\right[a,b]\times \mathbb{R}\times \mathbb{R},\mathbb{R})$. Throughout this paper, $\tau \in \mathcal{T}$.

Here, it is imperative to mention that the kernel structure in the definition of the Riemann–Liouville derivative is power–like, which is appropriate for nonlocal behavior. On the other hand, the Hadamard fractional derivative has a logarithmic-type kernel better suited for scaling invariance and physical phenomena containing long–range memory over logarithmic time scales. Thus, the underlying idea in the present research is to investigate a Langevin-type system containing the coupling of Riemann–Liouville and Hadamard fractional derivative operators, which are of different nature in terms of their kernels.

Our objective in this study is to establish the existence, uniqueness and Ulam–Hyers stability results for the proposed problem (1) and (2). We illustrate the obtained results with numerical examples.

In the next section, we present the related definitions and an auxiliary lemma. In Section 3, we prove the existence and uniqueness of solutions for the given problem by applying Leray–Schauder’s alternative and Banach’s fixed-point theorem, respectively. Section 4 contains the Ulam–Hyers stability results for problem (1) and (2). Numerical examples are presented in Section 5, and the graphical solutions are discussed in Section 6. We discuss some special cases of the obtained results in the last section.

2. A Preliminary Result

Let us begin this section with some related material from fractional calculus [12].

Definition 1. 

The (left) Riemann–Liouville fractional integral of order $\alpha \in {\mathbb{R}}^{+}$ for a function $\phi \in L^1[a,b]$, denoted by ${}^{RL}I_{a^{+}}^{\alpha }\phi$, is defined by

$${}^{RL}I_{a^{+}}^{\alpha }\phi \left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{a^{+}}{\overset{\tau }{\int }}{\left(\tau -z\right)}^{\alpha -1}\phi \left(z\right)dz,$$

where Γ denotes the Euler gamma function.

Definition 2. 

Let $\phi ,{\phi }^{\left(m\right)}\in L^1[a,b],a,b\in \mathbb{R}$ and $q\in (m-1,m),m\in \mathcal{N}$. The Riemann–Liouville fractional derivative of order q, denoted by ${}^{RL}D_{a^{+}}^q\phi$, is defined as

$${}^{RL}D_{a^{+}}^q\phi \left(\tau \right)={\displaystyle \frac{d^m}{d{\tau }^m}}{I}_{a^{+}}^{m-q}\phi \left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(m-q\right)}}{\displaystyle \frac{d^m}{d{\tau }^m}}\underset{a}{\overset{\tau }{\int }}{\left(\tau -z\right)}^{m-1-q}\phi \left(z\right)dz.$$

Definition 3. 

The Hadamard fractional integral of order $q>0$ for a continuous function $\phi :[a,\infty )\to \mathbb{R}$ is given by

$${}^{H}I_{a^{+}}^q\phi \left(\tau \right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}\underset{a}{\overset{\tau }{\int }}{\left(\log {\displaystyle \frac{\tau }{z}}\right)}^{q-1}{\displaystyle \frac{\phi \left(z\right)}{z}}dz,\tau >a>0,$$

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

Definition 4. 

The Hadamard fractional derivative of order $q>0$ for a continuous function $\phi :[a,\infty )\to \mathbb{R}$ is given by

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

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

Lemma 1. 

If φ is a continuous function and $p,q\in {\mathbb{R}}^{+}$, then

$\left(i\right)$

${}^{RL}I_{a^{+}}^p{}^{RL}I_{a^{+}}^q\phi \left(\tau \right)={}^{RL}I_{a^{+}}^{p+q}\phi \left(\tau \right)$ holds at almost every point $\tau \in \mathcal{T}$. In case $p+q>1$, then this relation holds at any point $\tau \in \mathcal{T}$ (Lemma 2.3 in [12], p. 73).

$\left(ii\right)$

For $q>p>0$, the relation ${}^{RL}D_{a^{+}}^p{}^{RL}I^q\phi \left(\tau \right)=I^{q-p}\phi \left(\tau \right)$ holds almost everywhere on $\mathcal{T}$ (Property 2.2 in [12]; p. 74).

Note that ${}^{RL}D_{a^{+}}^p{(\tau -a)}^{p-i}=0,i=1,2,\dots ,\left[p\right]+1$, where $\left[p\right]$ is the largest integer less than p and

$${}^{RL}D_{a^{+}}^p{(\tau -a)}^{\lambda }={\displaystyle \frac{\Gamma (\lambda +1)}{\Gamma (\lambda -p+1)}}{(\tau -a)}^{\lambda -p},p\ge 0,\lambda >-1$$

(Property 2.1 in [12]; p. 71).

Lemma 2. 

For $0<a<\infty$ and $p>0,q>0$, we have

1. 

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

2. 

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

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

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

while for $i=\left[p\right]+1$, we have

$$\left({}^{H}D_{a^{+}}^p{\left(\log {\displaystyle \frac{x}{a}}\right)}^{p-i}\right)\left(\tau \right)=0.$$

Theorem 1. 

For $g\in L^1(a,b)$ and $\left({}^{RL}I_{a^{+}}^{n-\alpha }g\right)\left(\tau \right)\in A{C}^n[a,b]$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \left({}^{RL}I_{a^{+}}^{\alpha }{}^{RL}D_{a^{+}}^{\alpha }g\right)\left(\tau \right)}& =& g\left(\tau \right)-\sum _{j=1}^n{\displaystyle \frac{\left({}^{RL}D_{a^{+}}^{\alpha -j}g\right)\left(a\right)}{\Gamma (\alpha -j+1)}}{(\tau -a)}^{\alpha -j},\hfill \end{array}$$

where $n=\left[\alpha \right]+1,\alpha >0$ and $A{C}^n[a,b]=\{f:[a,b]\to \mathbb{C}:D^{n-1}\left[f\left(\tau \right)\right]\in AC[a,b],D={\displaystyle \frac{d}{d\tau }}\}$.

Theorem 2. 

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

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

where $0<a<b<\infty$, and $n=\left[\alpha \right]+1,\alpha >0$ and $A{C}_{\delta }^n[a,b]=\{f:[a,b]\to \mathbb{R}:{\delta }^{n-1}\left[f\left(\tau \right)\right]\in AC[a,b],\delta =\tau {\displaystyle \frac{d}{d\tau }}\}$.

Here, we use the following notation:

$${}^{RL}I_{a^{+}}^q={}^{RL}I^q,{}^{RL}D_{a^{+}}^q={}^{RL}D^q,{}^{H}I_{a^{+}}^q={}^{H}I^q\mathrm{and}{}^{H}D_{a^{+}}^q={}^{H}D^q.$$

Now, we present the auxiliary lemma dealing with the linear version of the system (1) with boundary conditions (2).

Lemma 3. 

For ${\chi }_1,{\chi }_2\in C(\mathcal{T},\mathbb{R})$, $\Delta =(A_1{B}_2-A_2{B}_1)\ne 0$, the integral representation for the unique solution of the linear system

$${}^{RL}D^{{\rho }_1}\left({}^{RL}D^{{\omega }_1}+{\mu }_1\left(\tau \right)\right)x\left(\tau \right)={\chi }_1\left(\tau \right),{}^{H}D^{{\rho }_2}\left({}^{H}D^{{\omega }_2}+{\mu }_2\left(\tau \right)\right)y\left(\tau \right)={\chi }_2\left(\tau \right),$$

(3)

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

$$\begin{array}{ccc}\hfill {\displaystyle x\left(\tau \right)}& =& {\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(s\right)-{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(u\right)}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)y\left(u\right)\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(s\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(s\right)-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(u\right)-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)x\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(s\right)-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(s\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\}\hfill \end{array}$$

(4)

and

$$\begin{array}{ccc}\hfill {\displaystyle y\left(\tau \right)}& =& {\int }_a^{\tau }\left[{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(s\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(u\right)}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)y\left(u\right)\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(s\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(s\right)-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(u\right)-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)x\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}{\chi }_1\left(s\right)-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{{\chi }_2\left(s\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\},\hfill \end{array}$$

(5)

where

$$\begin{array}{ccc}\hfill & & {\mathcal{S}}_1\left(\tau \right)={\displaystyle \frac{B_2\Gamma \left({\rho }_1\right){(\tau -a)}^{{\rho }_1+{\omega }_1-1}}{\Delta \Gamma ({\rho }_1+{\omega }_1)}},{\mathcal{S}}_2\left(\tau \right)={\displaystyle \frac{A_2\Gamma \left({\rho }_1\right){(\tau -a)}^{{\rho }_1+{\omega }_1-1}}{\Delta \Gamma ({\rho }_1+{\omega }_1)}},\hfill \\ & & {\mathcal{Z}}_1\left(\tau \right)={\displaystyle \frac{B_1\Gamma \left({\rho }_2\right){\left(\log \frac{\tau }{a}\right)}^{{\rho }_2+{\omega }_2-1}}{\Delta \Gamma ({\rho }_2+{\omega }_2)}},{\mathcal{Z}}_2\left(\tau \right)={\displaystyle \frac{A_1\Gamma \left({\rho }_2\right){\left(\log \frac{\tau }{a}\right)}^{{\rho }_2+{\omega }_2-1}}{\Delta \Gamma ({\rho }_2+{\omega }_2)}},\hfill \\ & & A_1={\displaystyle \frac{\Gamma \left({\rho }_1\right){(b-a)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}},A_2={\displaystyle \frac{{\lambda }_1\Gamma \left({\rho }_2\right){\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}}+\sum _{i=1}^p{\displaystyle \frac{{\eta }_i\Gamma \left({\rho }_2\right){\left(\log \frac{{\alpha }_i}{a}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}},\hfill \\ & & B_1={\displaystyle \frac{{\lambda }_2\Gamma \left({\rho }_1\right){(b-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}+\sum _{j=1}^q{\displaystyle \frac{{\zeta }_j\Gamma \left({\rho }_1\right){({\sigma }_j-a)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}},B_2={\displaystyle \frac{\Gamma \left({\rho }_2\right){\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}},\hfill \\ & & {\Omega }_1={\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(s\right)-{}^{H}I^{{\omega }_2}\left({\mu }_2\left(s\right)y\left(s\right)\right)\right]ds\hfill \\ & & +\sum _{i=1}^p{\eta }_i\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left({\alpha }_i\right)-{}^{H}I^{{\omega }_2}\left({\mu }_2\left({\alpha }_i\right)y\left({\alpha }_i\right)\right)\right]-\left[{}^{RL}I^{I^{{\rho }_1+{\omega }_1}}{\chi }_1\left(b\right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left(b\right)x\left(b\right)\right)\right],\hfill \\ & & {\Omega }_2={\int }_a^b{\lambda }_2\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(s\right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ & & +\sum _{j=1}^q{\zeta }_j\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left({\sigma }_j\right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left({\sigma }_j\right)x\left({\sigma }_j\right)\right)\right]-\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(b\right){-}^H{I}^{{\omega }_2}\left({\mu }_2\left(b\right)y\left(b\right)\right)\right].\hfill \end{array}$$

(6)

Proof. 

By operating the Riemann–Liouville and the Hadamard integral operators ${}^{RL}I^{{\rho }_1}$ and ${}^{H}I^{{\rho }_2}$ on the first and second equations in (3), respectively, we get

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle ({}^{RL}D^{{\omega }_1}+{\mu }_1\left(\tau \right))x\left(\tau \right)={}^{RL}I^{{\rho }_1}{\chi }_1\left(\tau \right)+c_0{(\tau -a)}^{{\rho }_1-1},}\hfill \\ {\displaystyle ({}^{H}D^{{\omega }_2}+{\mu }_2\left(\tau \right))y\left(\tau \right)={}^{H}I^{{\rho }_2}{\chi }_2\left(\tau \right)+e_0{\left(\log \frac{\tau }{a}\right)}^{{\rho }_2-1},}\hfill \end{array}\end{array}$$

(7)

where $c_0,e_0\in \mathbb{R}$ are unknown arbitrary constants. Next, we apply the Riemann–Liouville and Hadamard integral operators ${}^{RL}I^{{\omega }_1}$ and ${}^{H}I^{{\omega }_2}$, respectively, on the first and second equations in (7) to obtain

$$\begin{array}{c}\hfill \begin{array}{c}x\left(\tau \right)={}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(\tau \right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)+c_0{\displaystyle \frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+{\omega }_1)}}{(\tau -a)}^{{\rho }_1+{\omega }_1-1}\hfill \\ +c_1{(\tau -a)}^{{\omega }_1-1}+c_2{(\tau -a)}^{{\omega }_1-2}+\cdots +c_n{(\tau -a)}^{{\omega }_1-n},\hfill \\ y\left(\tau \right)={}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(\tau \right)-{}^{H}I^{{\omega }_2}\left({\mu }_2\left(\tau \right)y\left(\tau \right)\right)+e_0{\displaystyle \frac{\Gamma \left({\rho }_2\right)}{\Gamma ({\rho }_2+{\omega }_2)}}{\left(\log {\displaystyle \frac{\tau }{a}}\right)}^{{\rho }_2+{\omega }_2-1}\hfill \\ +e_1{\left(\log {\displaystyle \frac{\tau }{a}}\right)}^{{\omega }_2-1}+e_2{\left(\log {\displaystyle \frac{\tau }{a}}\right)}^{{\omega }_2-2}+\cdots +e_m{\left(\log {\displaystyle \frac{\tau }{a}}\right)}^{{\omega }_2-m},\hfill \end{array}\end{array}$$

(8)

where $c_i,e_j\in \mathbb{R}$, $i=1,2,\dots ,n,j=1,2,\dots ,m$ are unknown arbitrary constants. Combining the conditions $x\left(a\right)=0=y\left(a\right)$ with (8), we find that $c_n=e_m=0$ as $n-1<{\omega }_1<n$ and $m-1<{\omega }_2<m$. Thus, (8) takes the form

$$\begin{array}{c}\hfill \begin{array}{c}x\left(\tau \right)={}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(\tau \right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)+c_0{\displaystyle \frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+{\omega }_1)}}{(\tau -a)}^{{\rho }_1+{\omega }_1-1}\hfill \\ +c_1{(\tau -a)}^{{\omega }_1-1}+c_2{(\tau -a)}^{{\omega }_1-2}+\cdots +c_{n-1}{(\tau -a)}^{{\omega }_1-(n-1)},\hfill \\ y\left(\tau \right)={}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(\tau \right)-{}^{H}I^{{\omega }_2}\left({\mu }_2\left(\tau \right)y\left(\tau \right)\right)+e_0{\displaystyle \frac{\Gamma \left({\rho }_2\right)}{\Gamma ({\rho }_2+{\omega }_2)}}{\left(\log {\displaystyle \frac{\tau }{a}}\right)}^{{\rho }_2+{\omega }_2-1}\hfill \\ {\displaystyle +e_1{\left(\log \frac{\tau }{a}\right)}^{{\omega }_2-1}+e_2{\left(\log \frac{\tau }{a}\right)}^{{\omega }_2-2}+\cdots +e_{m-1}{\left(\log \frac{\tau }{a}\right)}^{{\omega }_2-(m-1)}.}\hfill \end{array}\end{array}$$

(9)

From the first equation of (9), we have

$$\begin{array}{ccc}\hfill {}^{RL}D^{{\omega }_1-1}x\left(\tau \right)& =& {}^{RL}I^{{\rho }_1+1}{\chi }_1\left(\tau \right)-{}^{RL}I^1\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)+c_0{\displaystyle \frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+1)}}{(\tau -a)}^{{\rho }_1}+c_1\Gamma \left({\omega }_1\right),\hfill \\ \hfill {}^{RL}D^{{\omega }_1-2}x\left(\tau \right)& =& {}^{RL}I^{{\rho }_1+2}{\chi }_1\left(\tau \right)-{}^{RL}I^2\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)+c_0{\displaystyle \frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+2)}}{(\tau -a)}^{{\rho }_1+1}\hfill \\ & & +c_1\Gamma \left({\omega }_1\right)(\tau -a)+c_2\Gamma ({\omega }_1-1),\hfill \\ & & ⋮\hfill \\ \hfill {}^{RL}D^{{\omega }_1-(n-1)}x\left(\tau \right)& =& {}^{RL}I^{{\rho }_1+n-1}{\chi }_1\left(\tau \right)-{}^{RL}I^{n-1}\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)\hfill \\ & & {\displaystyle +c_0\frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+n-1)}{(\tau -a)}^{{\rho }_1+n-2}}\hfill \\ & & {\displaystyle +c_1\frac{\Gamma \left({\omega }_1\right)}{\Gamma (n-1)}{(\tau -a)}^{n-2}+\cdots +c_{n-1}\Gamma ({\omega }_1-n+2).}\hfill \end{array}$$

Using the above expressions in the conditions ${}^{RL}D^{{\omega }_1-v_1}x\left(a\right)=0$, $v_1=1,2,3,\dots ,n-1$, we find that $c_i=0$ for $i=1,2,\dots ,n-1$. Similarly, from the second equation of (9) with conditions ${}^{H}D^{{\omega }_2-v_2}y\left(a\right)=0$, $v_2=1,2,3,\dots ,m-1$, we can find that $e_j=0$ for $j=1,2,\dots ,m-1$. Now, inserting the above values of $c_i=0$ and $e_j=0$ in (9), we get

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle x\left(\tau \right)={}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(\tau \right)-{}^{RL}I^{{\omega }_1}\left({\mu }_1\left(\tau \right)x\left(\tau \right)\right)+c_0\frac{\Gamma \left({\rho }_1\right)}{\Gamma ({\rho }_1+{\omega }_1)}{(\tau -a)}^{{\rho }_1+{\omega }_1-1},}\hfill \\ {\displaystyle y\left(\tau \right)={}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(\tau \right)-{}^{H}I^{{\omega }_2}\left({\mu }_2\left(\tau \right)y\left(\tau \right)\right)+e_0\frac{\Gamma \left({\rho }_2\right)}{\Gamma ({\rho }_2+{\omega }_2)}{\left(\log \frac{\tau }{a}\right)}^{{\rho }_2+{\omega }_2-1}.}\hfill \end{array}\end{array}$$

(10)

Substituting (10) into the last two conditions of (2), we obtain

$$\begin{array}{c}{\displaystyle A_1{c}_0-A_2{e}_0={\Omega }_1,-B_1{c}_0+B_2{e}_0={\Omega }_2,}\hfill \end{array}$$

(11)

where $A_1$, $A_2$, $B_1$, $B_2$, ${\Omega }_1$, and ${\Omega }_2$ are given in (6).

Solving the system (11) for $c_0$ and $e_0$, we find that

$$\begin{array}{ccc}\hfill & & c_0={\displaystyle \frac{1}{\Delta }}\left[B_2{\Omega }_1+A_2{\Omega }_2\right],e_0={\displaystyle \frac{1}{\Delta }}\left[B_1{\Omega }_1+A_1{\Omega }_2\right].\hfill \end{array}$$

Inserting the above values of $c_0$ and $e_0$ in (10) and using (6) yields the solution (4) and (5). We obtain the converse of the lemma by direct computation. □

Corollary 1. 

( Special case: ${\mu }_1\left(\tau \right)={\mu }_1=$ constant and ${\mu }_2\left(\tau \right)={\mu }_2=$ constant) For ${\chi }_1,{\chi }_2\in C(\mathcal{T},\mathbb{R})$, $\Delta =(A_1{B}_2-A_2{B}_1)\ne 0$, the unique solution of the linear system

$$\begin{array}{c}{\displaystyle {}^{RL}D^{{\rho }_1}({}^{RL}D^{{\omega }_1}+{\mu }_1)x\left(\tau \right)={\chi }_1\left(\tau \right),{}^{H}D^{{\rho }_2}({}^{H}D^{{\omega }_2}+{\mu }_2)y\left(\tau \right)={\chi }_2\left(\tau \right),}\hfill \end{array}$$

(12)

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

$$\begin{array}{ccc}\hfill x\left(\tau \right)& =& {}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(\tau \right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left(\tau \right)+{\mathcal{S}}_1\left(\tau \right)[{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(s\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left(s\right)\right]ds\hfill \\ & & +\sum _{i=1}^p{\eta }_i\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left({\alpha }_i\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left({\alpha }_i\right)\right]-\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(b\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left(b\right)\right]]\hfill \\ & & {\displaystyle +{\mathcal{S}}_2\left(\tau \right)[{\int }_a^b{\lambda }_2\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(s\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left(s\right)\right]ds}\hfill \\ & & {\displaystyle +\sum _{j=1}^q{\zeta }_j\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left({\sigma }_j\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left({\sigma }_j\right)\right]-\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(b\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left(b\right)\right]],}\hfill \\ \hfill {\displaystyle y\left(\tau \right)}& =& {}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(\tau \right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left(\tau \right)+{\mathcal{Z}}_1\left(\tau \right)[{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(s\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left(s\right)\right]ds\hfill \\ & & {\displaystyle +\sum _{i=1}^p{\eta }_i\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left({\alpha }_i\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left({\alpha }_i\right)\right]-\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(b\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left(b\right)\right]]}\hfill \\ & & {\displaystyle +{\mathcal{Z}}_2\left(\tau \right)[{\int }_a^b{\lambda }_2\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left(s\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left(s\right)\right]ds}\hfill \\ & & {\displaystyle +\sum _{j=1}^q{\zeta }_j\left[{}^{RL}I^{{\rho }_1+{\omega }_1}{\chi }_1\left({\sigma }_j\right)-{\mu }_1{}^{RL}I^{{\omega }_1}x\left({\sigma }_j\right)\right]-\left[{}^{H}I^{{\rho }_2+{\omega }_2}{\chi }_2\left(b\right)-{\mu }_2{}^{H}I^{{\omega }_2}y\left(b\right)\right]].}\hfill \end{array}$$

(13)

3. Existence and Uniqueness Results

Let $\mathcal{H}$ denote the Banach space of all continuous functions from $[a,b]\to \mathbb{R}$ endowed with the supremum norm $\parallel x\parallel ={\sup }_{\tau \in \mathcal{T}}|x\left(\tau \right)|$. Then, the product space $\mathcal{H}\times \mathcal{H}$ is also a Banach space endowed with the norm $\parallel (x,y)\parallel =\parallel x\parallel +\parallel y\parallel ,(x,y)\in \mathcal{H}\times \mathcal{H}$.

By Lemma 3, problem (1) and (2) can be transformed into a fixed-point problem as $(x,y)=\mathcal{W}(x,y)$, where $\mathcal{W}:\mathcal{H}\times \mathcal{H}\to \mathcal{H}\times \mathcal{H}$ is an operator defined by

$$\mathcal{W}(x,y)\left(\tau \right)=\left(\begin{array}{c}{\mathcal{W}}_1(x,y)\left(\tau \right)\\ {\mathcal{W}}_2(x,y)\left(\tau \right)\end{array}\right),$$

(14)

where

$$\begin{array}{ccc}\hfill {\mathcal{W}}_1(x,y)\left(\tau \right)& =& {\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(u,x(u),y(u\left)\right)}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)y\left(u\right)\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,x(s),y(s\left)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(u,x\left(u\right),y\left(u\right))-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)x\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,x(s),y(s\left)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\mathcal{W}}_2(x,y)\left(\tau \right)& =& {\int }_a^{\tau }\left[{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,x(s),y(s\left)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(u,x(u),y(u\left)\right)}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)y\left(u\right)\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,x(s),y(s\left)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(u,x\left(u\right),y\left(u\right))-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)x\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,x(s),y(s\left)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)y\left(s\right)\right)}{s}}\right]ds\},\hfill \end{array}$$

(16)

Observe that the fixed points of the operator $\mathcal{W}$ are solutions to problem (1) and (2).

In the sequel, we set the notation

$$\begin{array}{ccc}\hfill {\displaystyle {\mathcal{U}}_1}& =& {\displaystyle \frac{{(b-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}\left[1+{\tilde{\mathcal{S}}}_1\right]+{\tilde{\mathcal{S}}}_2[{\displaystyle \frac{{\lambda }_2{(b-a)}^{{\rho }_1+{\omega }_1+1}}{\Gamma ({\rho }_1+{\omega }_1+2)}}+\sum _{j=1}^q|{\zeta }_j|{\displaystyle \frac{{({\sigma }_j-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}],\hfill \\ \hfill {\displaystyle {\mathcal{U}}_2}& =& {\tilde{\mathcal{S}}}_1\left[{\displaystyle \frac{{\lambda }_1{\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2+1}}{\Gamma ({\rho }_2+{\omega }_2+2)}}+\sum _{i=1}^p|{\eta }_i|{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}}\right]+{\tilde{\mathcal{S}}}_2{\displaystyle \frac{{\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}},\hfill \\ \hfill {\displaystyle {\mathcal{U}}_3}& =& \left[1+{\tilde{\mathcal{S}}}_1\right]{}^{RL}I^{{\omega }_1}|{\mu }_1\left(b\right)|+{\tilde{\mathcal{S}}}_2\left[{\int }_a^b{\lambda }_2{}^{RL}I^{{\omega }_1}|{\mu }_1\left(s\right)|ds+\sum _{j=1}^q|{\zeta }_j|{}^{RL}I^{{\omega }_1}|{\mu }_1\left({\sigma }_j\right)|\right],\hfill \\ \hfill {\displaystyle {\mathcal{U}}_4}& =& {\tilde{\mathcal{S}}}_1\left[{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{}^{H}I^{{\omega }_2}|{\mu }_2\left(s\right)|ds+\sum _{i=1}^p|{\eta }_i|{}^{H}I^{{\omega }_2}|{\mu }_2\left({\alpha }_i\right)|\right]+{\tilde{\mathcal{S}}}_2{}^{H}I^{{\omega }_2}|{\mu }_2\left(b\right)|,\hfill \\ \hfill {\displaystyle {\mathcal{V}}_1}& =& {\tilde{\mathcal{Z}}}_1{\displaystyle \frac{{(b-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}+{\tilde{\mathcal{Z}}}_2[{\lambda }_2{\displaystyle \frac{{(b-a)}^{{\rho }_1+{\omega }_1+1}}{\Gamma ({\rho }_1+{\omega }_1+2)}}+\sum _{j=1}^q|{\zeta }_j|{\displaystyle \frac{{({\sigma }_j-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}],\hfill \\ \hfill {\displaystyle {\mathcal{V}}_2}& =& {\tilde{\mathcal{Z}}}_1\left[{\displaystyle \frac{{\lambda }_1{\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2+1}}{\Gamma ({\rho }_2+{\omega }_2+2)}}+\sum _{i=1}^p|{\eta }_i|{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}}\right]+{\displaystyle \frac{{\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}}\left[1+{\tilde{\mathcal{Z}}}_2\right],\hfill \\ \hfill {\displaystyle {\mathcal{V}}_3}& =& {\tilde{\mathcal{Z}}}_1{}^{RL}I^{{\omega }_1}|{\mu }_1\left(b\right)|+{\tilde{\mathcal{Z}}}_2\left[{\int }_a^b{\lambda }_2{}^{RL}I^{{\omega }_1}|{\mu }_1\left(s\right)|ds+\sum _{j=1}^q|{\zeta }_j|{}^{RL}I^{{\omega }_1}|{\mu }_1\left({\sigma }_j\right)|\right],\hfill \\ \hfill {\displaystyle {\mathcal{V}}_4}& =& {\tilde{\mathcal{Z}}}_1\left[{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{}^{H}I^{{\omega }_2}|{\mu }_2\left(s\right)|ds+\sum _{i=1}^p|{\eta }_i|{}^{H}I^{{\omega }_2}|{\mu }_2\left({\alpha }_i\right)|\right]+\left[1+{\tilde{\mathcal{Z}}}_2\right]{}^{H}I^{{\omega }_2}|{\mu }_2\left(b\right)|,\hfill \end{array}$$

(17)

where

$${\tilde{\mathcal{S}}}_{\psi }=\underset{\tau \in \mathcal{T}}{\sup }|{\mathcal{S}}_{\psi }\left(\tau \right)|,{\tilde{\mathcal{Z}}}_{\psi }=\underset{\tau \in \mathcal{T}}{\sup }{|\mathcal{Z}}_{\psi }\left(\tau \right)|,\psi =1,2,3,4.$$

Remark 1. 

In the special case of ${\mu }_1\left(\tau \right)={\mu }_1=$ constant and ${\mu }_2\left(\tau \right)={\mu }_2=$ constant, ${\mathcal{U}}_3,{\mathcal{U}}_4,{\mathcal{V}}_3$ and ${\mathcal{V}}_4$ become

$$\begin{array}{ccc}\hfill {\displaystyle {\widehat{\mathcal{U}}}_3}& =& {\mu }_1[{\displaystyle \frac{{(b-a)}^{{\omega }_1}}{\Gamma ({\omega }_1+1)}}\left[1+{\tilde{\mathcal{S}}}_1\right]+{\tilde{\mathcal{S}}}_2[{\displaystyle \frac{{\lambda }_2{(b-a)}^{{\omega }_1+1}}{\Gamma ({\omega }_1+2)}}+\sum _{j=1}^q|{\zeta }_j|{\displaystyle \frac{{({\sigma }_j-a)}^{{\omega }_1}}{\Gamma ({\omega }_1+1)}},\hfill \\ \hfill {\displaystyle {\widehat{\mathcal{U}}}_4}& =& {\mu }_2\left[{\tilde{\mathcal{S}}}_1\left[{\lambda }_1{\displaystyle \frac{{\left(\log \frac{b}{a}\right)}^{{\omega }_2+1}}{\Gamma ({\omega }_2+2)}}+\sum _{i=1}^p|{\eta }_i|{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{a}\right)}^{{\omega }_2}}{\Gamma ({\omega }_2+1)}}\right]+{\tilde{\mathcal{S}}}_2{\displaystyle \frac{{\left(\log \frac{b}{a}\right)}^{{\omega }_2}}{\Gamma ({\omega }_2+1)}}\right],\hfill \\ \hfill {\widehat{\mathcal{V}}}_3& =& {\mu }_1[{\tilde{\mathcal{Z}}}_1{\displaystyle \frac{{(b-a)}^{{\omega }_1}}{\Gamma ({\omega }_1+1)}}+{\tilde{\mathcal{Z}}}_2[{\displaystyle \frac{{\lambda }_2{(b-a)}^{{\omega }_1+1}}{\Gamma ({\omega }_1+2)}}+\sum _{j=1}^q|{\zeta }_j|{\displaystyle \frac{{({\sigma }_j-a)}^{{\omega }_1}}{\Gamma ({\omega }_1+1)}},\hfill \\ \hfill {\displaystyle {\widehat{\mathcal{V}}}_4}& =& {\mu }_2\left[{\tilde{\mathcal{Z}}}_1\left[{\displaystyle \frac{{\lambda }_1{\left(\log \frac{b}{a}\right)}^{{\omega }_2+1}}{\Gamma ({\omega }_2+2)}}+\sum _{i=1}^p|{\eta }_i|{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{a}\right)}^{{\omega }_2}}{\Gamma ({\omega }_2+1)}}\right]+{\displaystyle \frac{{\left(\log \frac{b}{a}\right)}^{{\omega }_2}}{\Gamma ({\omega }_2+1)}}\left[1+{\tilde{\mathcal{Z}}}_2\right]\right].\hfill \end{array}$$

Now, we proceed to present our main results. In our first result, we establish the existence of solutions for problem (1) and (2), which rely on the Leray–Schauder alternative [39].

Theorem 3. 

Let $f,g:[a,b]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a continuous functions satisfying the condition

$\left(H_1\right)$

There exist real constants ${\gamma }_{\psi },{\delta }_{\psi }\ge 0,\psi =1,2$, and ${\gamma }_0,{\delta }_0>0$ such that

$$|f(\tau ,x,y)|\le {\gamma }_0+{\gamma }_1|x|+{\gamma }_2|y|,|g(\tau ,x,y)|\le {\delta }_0+{\delta }_1|x|+{\delta }_2|y|,\forall x,y\in \mathbb{R}.$$

Then, problem (1) and (2) has at least one solution on $[a,b]$, provided that $0<\max \{{\mathcal{F}}_1,{\mathcal{F}}_2\}<1$, where

$$\begin{array}{c}\hfill \begin{array}{c}{\mathcal{F}}_1={\gamma }_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\delta }_1({\mathcal{U}}_2+{\mathcal{V}}_2)+({\mathcal{U}}_3+{\mathcal{V}}_3),\hfill \\ {\mathcal{F}}_2={\gamma }_2({\mathcal{U}}_1+{\mathcal{V}}_1)+{\delta }_2({\mathcal{U}}_2+{\mathcal{V}}_2)+({\mathcal{U}}_4+{\mathcal{V}}_4),\hfill \end{array}\end{array}$$

(18)

and ${\mathcal{U}}_{\psi }$, ${\mathcal{V}}_{\psi }$, and $\psi =1,2,3,4$ are given in (17).

Proof. 

Let us first establish that the operator $\mathcal{W}:\mathcal{H}\times \mathcal{H}\to \mathcal{H}\times \mathcal{H}$ defined by (14) is completely continuous. Observe that the operator $\mathcal{W}$ is continuous in view of the continuity of functions f and g. Let ${\mathcal{P}}_{\nu }=\{(x,y)\in \mathcal{H}\times \mathcal{H}:\parallel (x,y)\parallel \le \nu \}.$

For any $(x,y)\in {\mathcal{P}}_{\nu }$, we have

$$\begin{array}{c}\hfill \begin{array}{c}|f(\tau ,x,y)|\le {\gamma }_0+{\gamma }_1\parallel x\parallel +{\gamma }_2\parallel y\parallel \le {\gamma }_0+({\gamma }_1+{\gamma }_2)\parallel (x,y)\parallel \le {\gamma }_0+({\gamma }_1+{\gamma }_2)\nu :={\mathcal{L}}_1,\hfill \\ |g(\tau ,x,y)|\le {\delta }_0+{\delta }_1\parallel x\parallel +{\delta }_2\parallel y\parallel \le {\delta }_0+({\delta }_1+{\delta }_2)\parallel (x,y)\parallel \le {\delta }_0+({\delta }_1+{\delta }_2)\nu :={\mathcal{L}}_2.\hfill \end{array}\end{array}$$

(19)

Then, for any $(x,y)\in {\mathcal{P}}_{\nu }$, we obtain

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_1(x,y)\parallel & \le & \underset{\tau \in \mathcal{T}}{\sup }\{{\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x\left(s\right),y\left(s\right))|+{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_1\left(\tau \right)|\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(u,x\left(u\right),y\left(u\right))|}{u}}+{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(u\right)y\left(u\right)|}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p|{\eta }_i|{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,x\left(s\right),y\left(s\right))|}{s}}+{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x\left(s\right),y\left(s\right))|+{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_2\left(\tau \right)|\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(u,x\left(u\right),y\left(u\right))|+{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(u\right)x\left(u\right)|\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q|{\zeta }_j|{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x\left(s\right),y\left(s\right))|+{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,x\left(s\right),y\left(s\right))|}{s}}+{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds\left\}\right\}\hfill \\ & \le & {\mathcal{L}}_1{\mathcal{U}}_1+{\mathcal{L}}_2{\mathcal{U}}_2+({\mathcal{U}}_3+{\mathcal{U}}_4)\nu ,\hfill \end{array}$$

where ${\mathcal{U}}_{\psi },\psi =1,2,3,4$ are given in (17). Hence,

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_1(x,y)\parallel & \le & {\mathcal{L}}_1{\mathcal{U}}_1+{\mathcal{L}}_2{\mathcal{U}}_2+({\mathcal{U}}_3+{\mathcal{U}}_4)\nu .\hfill \end{array}$$

(20)

Similarly, we can find that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_2(x,y)\parallel & \le & {\mathcal{L}}_1{\mathcal{V}}_1+{\mathcal{L}}_2{\mathcal{V}}_2+({\mathcal{V}}_3+{\mathcal{V}}_4)\nu ,\hfill \end{array}$$

(21)

where ${\mathcal{V}}_{\psi },\psi =1,2,3,4$ are given in (17).

Hence,

$$\begin{array}{ccc}\hfill \parallel \mathcal{W}(x,y)\parallel & =& \parallel {\mathcal{W}}_1(x,y)\parallel +\parallel {\mathcal{W}}_2(x,y)\parallel \hfill \\ & \le & {\mathcal{L}}_1({\mathcal{U}}_1+{\mathcal{U}}_2)+{\mathcal{L}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)+({\mathcal{U}}_3+{\mathcal{U}}_4+{\mathcal{V}}_3+{\mathcal{V}}_4)\nu ,\hfill \end{array}$$

which shows that $\mathcal{W}\left({\mathcal{P}}_{\nu }\right)$ is uniformly bounded.

Next, we show that $\mathcal{W}\left({\mathcal{P}}_{\nu }\right)$ is equicontinuous. For that, we take ${\tau }_1,{\tau }_2\in \mathcal{T}$ with ${\tau }_1<{\tau }_2,(x,y)\in {\mathcal{P}}_{\nu }$. Then, we obtain

$$\begin{array}{ccc}& & |{\mathcal{W}}_1(x,y)\left({\tau }_2\right)-{\mathcal{W}}_1(x,y)\left({\tau }_1\right)|\hfill \\ & \le & |{\int }_a^{{\tau }_2}\left[{\displaystyle \frac{{({\tau }_2-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{({\tau }_2-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds\hfill \\ & & -{\int }_a^{{\tau }_1}\left[{\displaystyle \frac{{({\tau }_1-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,x\left(s\right),y\left(s\right))-{\displaystyle \frac{{({\tau }_1-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)x\left(s\right)\right)\right]ds|\hfill \\ & & +|{\mathcal{S}}_1\left({\tau }_2\right)-{\mathcal{S}}_1\left({\tau }_1\right)||{\tilde{\Omega }}_1|+|{\mathcal{S}}_2\left({\tau }_2\right)-{\mathcal{S}}_2\left({\tau }_1\right)||{\tilde{\Omega }}_2|\hfill \\ & \le & {\displaystyle \frac{{\mathcal{L}}_1}{\Gamma ({\rho }_1+{\omega }_1+1)}}\left[2{({\tau }_2-{\tau }_1)}^{{\rho }_1+{\omega }_1}+|{({\tau }_2-a)}^{{\rho }_1+{\omega }_1}-{({\tau }_1-a)}^{{\rho }_1+{\omega }_1}|\right]\hfill \\ & & +\nu |{\int }_a^{{\tau }_1}{\displaystyle \frac{{({\tau }_2-s)}^{{\omega }_1-1}-{({\tau }_1-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}{\mu }_1\left(s\right)ds+{\int }_{{\tau }_1}^{{\tau }_2}{\displaystyle \frac{{({\tau }_2-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}{\mu }_1\left(s\right)ds|\hfill \\ & & +\left[{\displaystyle \frac{|B_2|\Gamma \left({\rho }_1\right)}{\Delta \Gamma ({\rho }_1+{\omega }_1)}}|{({\tau }_2-a)}^{{\rho }_1+{\omega }_1-1}-{({\tau }_1-a)}^{{\rho }_1+{\omega }_1-1}|\right]|{\tilde{\Omega }}_1|\hfill \\ & & +\left[{\displaystyle \frac{|A_2|\Gamma \left({\rho }_1\right)}{\Delta \Gamma ({\rho }_1+{\omega }_1)}}|{({\tau }_2-a)}^{{\rho }_1+{\omega }_1-1}-{({\tau }_1-a)}^{{\rho }_1+{\omega }_1-1}|\right]|{\tilde{\Omega }}_2|\hfill \\ & & ⟶0\mathrm{as}({\tau }_2-{\tau }_1)\to 0\mathrm{independently}\mathrm{of}(x,y)\in {\mathcal{P}}_{\nu },\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill |{\tilde{\Omega }}_1|& \le & {\displaystyle \frac{{\mathcal{L}}_1}{\Gamma ({\rho }_1+{\omega }_1+1)}}{(b-a)}^{{\rho }_1+{\omega }_1}+{\mathcal{L}}_2\left[{\displaystyle \frac{{\lambda }_1{\left(\log \frac{b}{a}\right)}^{{\rho }_2+{\omega }_2+1}}{\Gamma ({\rho }_2+{\omega }_2+2)}}+\sum _{i=1}^p|{\eta }_i|{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{a}\right)}^{{\rho }_2+{\omega }_2}}{\Gamma ({\rho }_2+{\omega }_2+1)}}\right]\hfill \\ & & +\nu \left[{}^{RL}I^{{\omega }_1}|{\mu }_1\left(b\right)|+{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{}^{H}I^{{\omega }_2}|{\mu }_2\left(s\right)|ds+\sum _{i=1}^p|{\eta }_i|{}^{H}I^{{\omega }_2}|{\mu }_2\left({\alpha }_i\right)|\right]\hfill \\ \hfill |{\tilde{\Omega }}_2|& \le & {\mathcal{L}}_1[{\displaystyle \frac{{\lambda }_2{(b-a)}^{{\rho }_1+{\omega }_1+1}}{\Gamma ({\rho }_1+{\omega }_1+2)}}+\sum _{j=1}^q|{\zeta }_j|{\displaystyle \frac{{({\sigma }_j-a)}^{{\rho }_1+{\omega }_1}}{\Gamma ({\rho }_1+{\omega }_1+1)}}]+{\displaystyle \frac{{\mathcal{L}}_2}{\Gamma ({\rho }_2+{\omega }_2+1)}}{\left(\log {\displaystyle \frac{b}{a}}\right)}^{{\rho }_2+{\omega }_2}\hfill \\ & & +\nu \left[{\int }_a^b{\lambda }_2{}^{RL}I^{{\omega }_1}|{\mu }_1\left(s\right)|ds+\sum _{j=1}^q|{\zeta }_j|{}^{RL}I^{{\omega }_1}|{\mu }_1\left({\sigma }_j\right)|+{}^{H}I^{{\omega }_2}|{\mu }_2\left(b\right)|\right].\hfill \end{array}$$

Thus, ${\mathcal{W}}_1\left({\mathcal{P}}_{\nu }\right)$ and ${\mathcal{W}}_2\left({\mathcal{P}}_{\nu }\right)$ are equicontinuous. Consequently, $\mathcal{W}\left({\mathcal{P}}_{\nu }\right)$ is equicontinuous. Therefore, according to the Arzelà–Ascoli theorem, $\mathcal{W}\left({\mathcal{P}}_{\nu }\right)$ is completely continuous.

Next, we introduce a set $\mathcal{G}=\left\{\right(x,y)\in \mathcal{H}\times \mathcal{H}:(x,y)=\varrho \mathcal{W}(x,y),0<\varrho <1\}$ and verify that it is bounded. Letting $(x,y)\in \mathcal{G}$, we have $(x,y)=\varrho \mathcal{W}(x,y)$. Thus, $x\left(\tau \right)=\varrho {\mathcal{W}}_1(x,y)\left(\tau \right)$ and $y\left(\tau \right)=\varrho {\mathcal{W}}_2(x,y)\left(\tau \right)$ for $\tau \in \mathcal{T}$. Then, it follows by the assumption $\left(H_1\right)$ that

$$\begin{array}{ccc}\hfill \parallel x\parallel & =& \underset{\tau \in \mathcal{T}}{\sup }|x\left(\tau \right)|\le \underset{\tau \in \mathcal{T}}{\sup }|{\mathcal{W}}_1(x,y)\left(\tau \right)|\hfill \\ & \le & \underset{\tau \in \mathcal{T}}{\sup }\{{\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\gamma }_0+{\gamma }_1|x|+{\gamma }_2|y|\right)+{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_1\left(\tau \right)|\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\delta }_0+{\delta }_1|x|+{\delta }_2|y|\right)}{u}}+{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(u\right)y\left(u\right)|}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p|{\eta }_i|{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\delta }_0+{\delta }_1|x|+{\delta }_2|y|\right)}{s}}+{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\gamma }_0+{\gamma }_1|x|+{\gamma }_2|y|\right)+{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_2\left(\tau \right)|\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\gamma }_0+{\gamma }_1|x|+{\gamma }_2|y|\right)+{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(u\right)x\left(u\right)|\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q|{\zeta }_j|{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\gamma }_0+{\gamma }_1|x|+{\gamma }_2|y|\right)+{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\delta }_0+{\delta }_1|x|+{\delta }_2|y|\right)}{s}}+{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \parallel x\parallel & \le & ({\gamma }_0{\mathcal{U}}_1+{\delta }_0{\mathcal{U}}_2)+({\gamma }_1{\mathcal{U}}_1+{\delta }_1{\mathcal{U}}_2+{\mathcal{U}}_3)\parallel x\parallel +({\gamma }_2{\mathcal{U}}_1+{\delta }_2{\mathcal{U}}_2+{\mathcal{U}}_4)\parallel y\parallel .\hfill \end{array}$$

(22)

Similarly, one can find that

$$\begin{array}{ccc}\hfill \parallel y\parallel & \le & ({\gamma }_0{\mathcal{V}}_1+{\delta }_0{\mathcal{V}}_2)+({\gamma }_1{\mathcal{V}}_1+{\delta }_1{\mathcal{V}}_2+{\mathcal{V}}_3)\parallel x\parallel +({\gamma }_2{\mathcal{V}}_1+{\delta }_2{\mathcal{V}}_2+{\mathcal{V}}_4)\parallel y\parallel .\hfill \end{array}$$

(23)

Thus, from (22) and (23) together with the values of ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ given in (18), we have

$$\begin{array}{ccc}\hfill \parallel x\parallel +\parallel y\parallel & \le & {\gamma }_0({\mathcal{U}}_1+{\mathcal{V}}_1)+{\delta }_0({\mathcal{U}}_2+{\mathcal{V}}_2)+\max ({\mathcal{F}}_1,{\mathcal{F}}_2)\left(\parallel x\parallel +\parallel y\parallel \right),\hfill \end{array}$$

which can alternatively be written as

$$\begin{array}{ccc}\hfill \parallel (x,y)\parallel & \le & {\displaystyle \frac{1}{{\mathcal{Q}}_0}}[{\gamma }_0({\mathcal{U}}_1+{\mathcal{V}}_1)+{\delta }_0({\mathcal{U}}_2+{\mathcal{V}}_2)],\hfill \end{array}$$

where ${\mathcal{Q}}_0=1-\max ({\mathcal{F}}_1,{\mathcal{F}}_2)$.

Therefore, the set $\mathcal{G}$ is bounded. In consequence, with the Leray–Schauder alternative [39], we deduce that the operator $\mathcal{W}$ has at least one fixed point. Hence, problem (1) and (2) has at least one solution on $\mathcal{T}$. □

Next, we accomplish a uniqueness result for problem (1) and (2) with the aid of a fixed-point theorem due to Banach [39].

Theorem 4. 

Assume that $f,g:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the Lipschitz conditions, as follows:

$\left(H_2\right)$

There exist positive constants ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ such that

$$\begin{array}{ccc}& & |f(\tau ,x_2,y_2)-f(\tau ,x_1,y_1)|\le {\mathcal{K}}_1(|x_2-x_1|+|y_2-y_1|),\hfill \\ & & |g(\tau ,x_2,y_2)-g(\tau ,x_1,y_1)|\le {\mathcal{K}}_2(|x_2-x_1|+|y_2-y_1|),\forall x_1,x_2,y_1,y_2\in \mathbb{R}.\hfill \end{array}$$

Then, problem (1) and (2) has a unique solution on $[a,b]$ if

$$\mathbb{H}={\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)+\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)<1,$$

(24)

where ${\mathcal{U}}_p$, ${\mathcal{V}}_p$, and $p=1,2,3,4$ are given in (17).

Proof. 

For verifying the hypotheses of Banach’s fixed-point theorem, we consider a closed ball ${\mathcal{A}}_r=\{(x,y)\in \mathcal{H}\times \mathcal{H}:\parallel (x,y)\parallel \le r\}$ with

$$r\ge {\displaystyle \frac{{\mathcal{B}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{B}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)}{1-\left[\mathcal{K}1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\right]-\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)}},$$

(25)

where ${\sup }_{\tau \in \mathcal{T}}|f(\tau ,0,0)|={\mathcal{B}}_1<\infty$ and ${\sup }_{\tau \in \mathcal{T}}|g(\tau ,0,0)|={\mathcal{B}}_2<\infty$. Now, we establish that $\mathcal{W}{\mathcal{A}}_r\subset {\mathcal{A}}_r$, where $\mathcal{W}:{\mathcal{A}}_r⟶\mathcal{H}\times \mathcal{H}$ is given by (14). By $\left(H_2\right)$, we have

$$\begin{array}{ccc}\hfill |f(\tau ,x\left(\tau \right),y\left(\tau \right))|& =& |f(\tau ,x\left(\tau \right),y\left(\tau \right))-f(\tau ,0,0)+f(\tau ,0,0)|\le {\mathcal{K}}_1(\parallel x\parallel +\parallel y\parallel )+{\mathcal{B}}_1\hfill \\ & \le & {\mathcal{K}}_{1}r+{\mathcal{B}}_1,\hfill \\ \hfill |g(\tau ,x\left(\tau \right),y\left(\tau \right))|& =& |g(\tau ,x\left(\tau \right),y\left(\tau \right))-g(\tau ,0,0)+g(\tau ,0,0)|\le {\mathcal{K}}_2(\parallel x\parallel +\parallel y\parallel )+{\mathcal{B}}_2\hfill \\ & \le & {\mathcal{K}}_{2}r+{\mathcal{B}}_2.\hfill \end{array}$$

(26)

For $(x,y)\in {\mathcal{A}}_r$, it follows by using (26) that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_1(x,y)\parallel & \le & \underset{\tau \in \mathcal{T}}{\sup }\{{\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right)+{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_1\left(\tau \right)|\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\mathcal{K}}_{2}r+{\mathcal{B}}_2\right)}{u}}+{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(u\right)y\left(u\right)|}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p|{\eta }_i|{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\mathcal{K}}_{2}r+{\mathcal{B}}_2\right)}{s}}+{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right)+{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\}\hfill \\ \hfill {\displaystyle }& & \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_2\left(\tau \right)|\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right)+{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(u\right)x\left(u\right)|\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q|{\zeta }_j|{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right)+{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)x\left(s\right)|\right]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left({\mathcal{K}}_{2}r+{\mathcal{B}}_2\right)}{s}}+{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)y\left(s\right)|}{s}}\right]ds\left\}\right\}\hfill \\ & \le & \left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right){\mathcal{U}}_1+{\mathcal{U}}_3\parallel x\parallel +\left({\mathcal{K}}_{2}r+{\mathcal{B}}_2\right){\mathcal{U}}_2+{\mathcal{U}}_4\parallel y\parallel .\hfill \end{array}$$

(27)

Likewise, we can find that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_2(x,y)\parallel & \le & \left({\mathcal{K}}_{1}r+{\mathcal{B}}_1\right){\mathcal{V}}_1+{\mathcal{V}}_3\parallel x\parallel +\left({\mathcal{K}}_{2}r+{\mathcal{B}}_2\right){\mathcal{V}}_2+{\mathcal{V}}_4\parallel y\parallel .\hfill \end{array}$$

(28)

From (28) and (28) together with (25), we get

$$\begin{array}{ccc}\hfill \parallel \mathcal{W}(x,y)\parallel & =& \parallel {\mathcal{W}}_1(x,y)\parallel +\parallel {\mathcal{W}}_2(x,y)\parallel \hfill \\ & \le & \left[{\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\right]r+{\mathcal{B}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{B}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\hfill \\ & & +\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)r\le r,\hfill \end{array}$$

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

Next, we verify that the operator $\mathcal{W}$ is a contraction. For that, let $(x_1,y_1),(x_2,y_2)\in {\mathcal{A}}_r$. Then, for any $\tau \in \mathcal{T}$, we obtain

$$\begin{array}{ccc}\hfill & & \parallel {\mathcal{W}}_1(x_2,y_2)-{\mathcal{W}}_1(x_1,y_1)\parallel \hfill \\ & \le & \underset{\tau \in \mathcal{T}}{\sup }\left\{{\int }_a^{\tau }\right[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x_2\left(s\right),y_2\left(s\right))-f(s,x_1\left(s\right),y_1\left(s\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)(x_2\left(s\right)-x_1\left(s\right))|]ds\hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_1\left(\tau \right)|\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(u,x_2\left(u\right),y_2\left(u\right))-g(u,x_1\left(u\right),y_1\left(u\right))|}{u}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(u\right)(y_2\left(u\right)-y_1\left(u\right))|}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p|{\eta }_i|{\int }_a^{{\alpha }_i}[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,x_2\left(s\right),y_2\left(s\right))-g(s,x_1\left(s\right),y_1\left(s\right))|}{s}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)(y_2\left(s\right)-y_1\left(s\right))|}{s}}]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x_2\left(s\right),y_2\left(s\right))-f(s,x_1\left(s\right),y_1\left(s\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)(x_2\left(s\right)-x_1\left(s\right))|\left]ds\right\}\hfill \\ \hfill {\displaystyle }& & +|{\mathcal{S}}_2\left(\tau \right)|\{{\int }_a^b{\int }_a^s{\lambda }_2[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(u,x_2\left(u\right),y_2\left(u\right))-f(u,x_1\left(u\right),y_1\left(u\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(u\right)(x_2\left(u\right)-x_1\left(u\right))|]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q|{\zeta }_j|{\int }_a^{{\sigma }_j}[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,x_2\left(s\right),y_2\left(s\right))-f(s,x_1\left(s\right),y_1\left(s\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)(x_2\left(s\right)-x_1\left(s\right))|]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,x_2\left(s\right),y_2\left(s\right))-g(s,x_1\left(s\right),y_1\left(s\right))|}{s}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)(y_2\left(s\right)-y_1\left(s\right))|}{s}}]ds,\hfill \end{array}$$

(29)

which implies that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_1(x_2,y_2)-{\mathcal{W}}_1(x_1,y_1)\parallel & \le & \left({\mathcal{K}}_1{\mathcal{U}}_1+{\mathcal{K}}_2{\mathcal{U}}_2\right)(\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel )\hfill \\ & & +{\mathcal{U}}_3\parallel x_2-x_1\parallel +{\mathcal{V}}_4\parallel y_2-y_1\parallel .\hfill \end{array}$$

Similarly, we can find that

$$\begin{array}{ccc}\hfill \parallel {\mathcal{W}}_2(x_2,y_2)-{\mathcal{W}}_2(x_1,y_1)\parallel & \le & \left({\mathcal{K}}_1{\mathcal{V}}_1+{\mathcal{K}}_2{\mathcal{V}}_2\right)(\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel )\hfill \\ & & +{\mathcal{U}}_3\parallel x_2-x_1\parallel +{\mathcal{U}}_4\parallel y_2-y_1\parallel .\hfill \end{array}$$

From the foregoing inequalities, we get

$$\begin{array}{ccc}\hfill \parallel \mathcal{W}(x_2,y_2)-\mathcal{W}(x_1,y_1)\parallel & \le & [{\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\hfill \\ & & +\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)\left]\right(\parallel x_2-x_1\parallel +\parallel y_2-y_1\parallel ),\hfill \end{array}$$

which, by (24), implies that $\mathcal{W}$ is a contraction. Therefore, there exists a unique fixed point for the operator $\mathcal{W}$ in ${\mathcal{A}}_r$ when applying Banach’s fixed–point theorem, which corresponds to a unique solution to problem (1) and (2) on $\mathcal{T}$. □

4. Stability Analysis

Let us first develop the arguments for the Ulam–Hyers stability [40] of problem (1) and (2).

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

$$\left\{\begin{array}{c}{\displaystyle |{}^{RL}D^{{\rho }_1}({}^{RL}D^{{\omega }_1}+{\mu }_1\left(\tau \right))\widehat{u}\left(\tau \right)-f(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))|\le {\epsilon }_1,}\hfill \\ {\displaystyle |{}^{H}D^{{\rho }_2}({}^{H}D^{{\omega }_2}+{\mu }_2\left(\tau \right))\widehat{v}\left(\tau \right)-g(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))|\le {\epsilon }_2.}\hfill \end{array}\right.$$

(30)

If $(\widehat{u},\widehat{v})\in \mathcal{H}\times \mathcal{H}$ is a solution of the system of inequalities (30) with boundary conditions (2), then there exist functions ${\varphi }_1,{\varphi }_2\in C(\mathcal{T},\mathbb{R})$ with $|{\varphi }_1\left(\tau \right)|\le {\epsilon }_1$, $|{\varphi }_2\left(\tau \right)|\le {\epsilon }_2$, and $\tau \in \mathcal{T}$, such that $(\widehat{u},\widehat{v})\in \mathcal{H}\times \mathcal{H}$ satisfies the system of Riemann–Liouville and Hadamard–type fractional Langevin equations

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D^{{\rho }_1}({}^{RL}D^{{\omega }_1}+{\mu }_1\left(\tau \right))\widehat{u}\left(\tau \right)=f(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))+{\varphi }_1\left(\tau \right),}\hfill \\ {\displaystyle {}^{H}D^{{\rho }_2}({}^{H}D^{{\omega }_2}+{\mu }_2\left(\tau \right))\widehat{v}\left(\tau \right)=g(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))+{\varphi }_2\left(\tau \right).}\hfill \end{array}\right.$$

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

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}I^{{\rho }_1}\left({}^{RL}D^{{\omega }_1}+{\mu }_1\left(\tau \right)\right)\widehat{u}\left(\tau \right)=f(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))+{\varphi }_1\left(\tau \right),}\hfill \\ {\displaystyle {}^{H}D^{{\rho }_2}\left({}^{H}D^{{\omega }_2}+{\mu }_2\left(\tau \right)\right)\widehat{v}\left(\tau \right)=g(\tau ,\widehat{u}\left(\tau \right),\widehat{v}\left(\tau \right))+{\varphi }_2\left(\tau \right),}\hfill \\ {\displaystyle \widehat{u}\left(a\right)=0,{}^{RL}D^{{\omega }_1-v_1}\widehat{u}\left(a\right)=0,v_1=1,2,3,\dots ,n-1,}\hfill \\ {\displaystyle \widehat{v}\left(a\right)=0,{}^{H}D^{{\omega }_2-v_2}\widehat{v}\left(a\right)=0,v_2=1,2,3,\dots ,m-1,}\hfill \\ {\displaystyle \widehat{u}\left(b\right)={\int }_a^b\frac{{\lambda }_1}{s}\widehat{v}\left(s\right)ds+\sum _{i=1}^p{\eta }_i\widehat{v}\left({\alpha }_i\right),{\alpha }_i\in (a,b),{\lambda }_1\ge 0,{\eta }_i\in \mathbb{R},}\hfill \\ {\displaystyle \widehat{v}\left(b\right)={\lambda }_2{\int }_a^b\widehat{u}\left(s\right)ds+\sum _{j=1}^q{\zeta }_j\widehat{u}\left({\sigma }_j\right),{\sigma }_j\in (a,b),{\lambda }_2\ge 0,{\zeta }_j\in \mathbb{R}.}\hfill \end{array}\right.$$

(31)

Definition 5. 

System (1) and (2) is called Ulam–Hyers–stable if we can find $c=(c_f,c_g)>0$, such that—for each solution $(\widehat{u},\widehat{v})\in \mathcal{H}\times \mathcal{H}$ of (31)—there exists a unique solution $(x,y)\in \mathcal{H}\times \mathcal{H}$ of system (1) and (2) satisfying

$$\parallel (\widehat{u},\widehat{v})-(x,y)\parallel \le c{({\epsilon }_1,{\epsilon }_2)}^T.$$

Definition 6. 

System (1) and (2) is said to be generalized Ulam–Hyers-stable if there exists a unique solution $(x,y)\in \mathcal{H}\times \mathcal{H}$ of system (1) and (2) satisfying

$$\parallel (\widehat{u},\widehat{v})-(x,y)\parallel \le \Lambda \left(ϵ\right),$$

for each solution $(\widehat{u},\widehat{v})\in \mathcal{H}\times \mathcal{H}$ of (31), where $\Lambda \in C({\mathbb{R}}^{+}\times {\mathbb{R}}^{+},{\mathbb{R}}^{+})$ with $\Lambda (0,0)=0$.

Theorem 5. 

Let $f,g\in C(\mathcal{T}\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ and the assumption $\left(H_2\right)$ and the condition (24) hold. Then, system (1) and (2) is Ulam–Hyers–stable and hence generalized Ulam–Hyers–stable in $\mathcal{H}\times \mathcal{H}$.

Proof. 

By Lemma 3, we can write the solution of (31) as

$$\begin{array}{ccc}& & \widehat{u}\left(\tau \right)\hfill \\ & =& {\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_1\left(s\right)\right)-{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_1\left(\tau \right)\left\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\right[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))+{\varphi }_2\left(u\right)\right)}{u}}\hfill \\ & & -{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)\widehat{v}\left(u\right)\right)}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_2\left(s\right)\right)}{s}}\hfill \\ \hfill {\displaystyle }& & -{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_1\left(s\right)\right)-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_2\left(\tau \right)\left\{{\int }_a^b{\int }_a^s{\lambda }_2\right[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))+{\varphi }_1\left(u\right)\right)\hfill \\ & & -{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)\widehat{u}\left(u\right)\right)]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_1\left(s\right)\right)\hfill \\ \hfill {\displaystyle }& & -{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_2\left(s\right)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\},\hfill \end{array}$$

$$\begin{array}{ccc}& & \widehat{v}\left(\tau \right)\hfill \\ & =& {\int }_a^{\tau }\left[{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_2\left(s\right)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_1\left(\tau \right)\left\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\right[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))+{\varphi }_2\left(u\right)\right)}{u}}\hfill \\ & & -{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)\widehat{v}\left(u\right)\right)}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_2\left(s\right)\right)}{s}}\hfill \\ \hfill {\displaystyle }& & -{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_1\left(s\right)\right)-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & +{\mathcal{Z}}_2\left(\tau \right)\left\{{\int }_a^b{\int }_a^s{\lambda }_2\right[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))+{\varphi }_1\left(u\right)\right)\hfill \\ & & -{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)\widehat{u}\left(u\right)\right)]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}\left(f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_1\left(s\right)\right)\hfill \\ \hfill {\displaystyle }& & -{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{\left(g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))+{\varphi }_2\left(s\right)\right)}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\}.\hfill \end{array}$$

In view of $|{\varphi }_1|<{\epsilon }_1$, $|{\varphi }_2|<{\epsilon }_2$ and (17), we have

$$\begin{array}{ccc}& & \underset{\tau \in \mathcal{T}}{\sup }|\widehat{u}\left(\tau \right)-{\int }_a^{\tau }\left[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\mathcal{S}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)\widehat{v}\left(u\right)\right)}{u}}\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & -{\mathcal{S}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_2\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)\widehat{u}\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\}|\le {\mathcal{U}}_1{\epsilon }_1+{\mathcal{U}}_2{\epsilon }_2\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\displaystyle }& & \underset{\tau \in \mathcal{T}}{\sup }|\widehat{v}\left(\tau \right)-{\int }_a^{\tau }\left[{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))}{s}}-{\displaystyle \frac{{\left(\log \frac{\tau }{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\mathcal{Z}}_1\left(\tau \right)\{{\int }_a^b{\displaystyle \frac{{\lambda }_1}{s}}{\int }_a^s\left[{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))}{u}}-{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(u\right)\widehat{v}\left(u\right)\right)}{u}}\right]duds\hfill \\ {\displaystyle }\hfill & & +\sum _{i=1}^p{\eta }_i{\int }_a^{{\alpha }_i}\left[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))}{s}}-{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\}\hfill \\ \hfill {\displaystyle }& & \hfill \\ \hfill {\displaystyle }& & -{\mathcal{Z}}_2\left(\tau \right)\{{\int }_a^b{\int }_a^s{\lambda }_1\left[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))-{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(u\right)\widehat{u}\left(u\right)\right)\right]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q{\zeta }_j{\int }_a^{{\sigma }_j}\left[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}\left({\mu }_1\left(s\right)\widehat{u}\left(s\right)\right)\right]ds\hfill \\ \hfill {\displaystyle }& & -{\int }_a^b\left[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))}{s}}-{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{\left({\mu }_2\left(s\right)\widehat{v}\left(s\right)\right)}{s}}\right]ds\}|\le {\mathcal{V}}_1{\epsilon }_1+{\mathcal{V}}_2{\epsilon }_2.\hfill \end{array}$$

By $\left(H_2\right)$, we have that

$$\begin{array}{ccc}\hfill \parallel \widehat{u}-x\parallel & =& \underset{\tau \in \mathcal{T}}{\sup }|\widehat{u}\left(\tau \right)-x\left(\tau \right)|\hfill \\ & \le & {\mathcal{U}}_1{\epsilon }_1+{\mathcal{U}}_2{\epsilon }_2+\underset{\tau \in \mathcal{T}}{\sup }\left\{{\int }_a^{\tau }\right[{\displaystyle \frac{{(\tau -s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-f(s,x\left(s\right),y\left(s\right))|\hfill \\ & & +{\displaystyle \frac{{(\tau -s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)\left(\widehat{u}\left(s\right)-x\left(s\right)\right)|]ds\hfill \\ & & {\displaystyle +{\mathcal{S}}_1\left(\tau \right)\left\{{\int }_a^b\frac{{\lambda }_1}{s}{\int }_a^s\right[\frac{{\left(\log \frac{s}{u}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}\frac{|g(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))-g(u,x\left(u\right),y\left(u\right))|}{u}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{s}{u}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(u\right)\left(\widehat{v}\left(u\right)-y\left(u\right)\right)|}{u}}]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{i=1}^p|{\eta }_i|{\int }_a^{{\alpha }_i}[{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-g(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{{\alpha }_i}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)\left(\widehat{v}\left(s\right)-y\left(s\right)\right)|}{s}}]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b[{\displaystyle \frac{{(b-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-f(s,x\left(s\right),y\left(s\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{(b-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)\left(\widehat{u}\left(s\right)-x\left(s\right)\right)|\left]ds\right\}\hfill \\ \hfill {\displaystyle }& & +{\mathcal{S}}_2\left(\tau \right)\left\{{\int }_a^b{\int }_a^s{\lambda }_1\right[{\displaystyle \frac{{(s-u)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(u,\widehat{u}\left(u\right),\widehat{v}\left(u\right))-f(u,x\left(u\right),y\left(u\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{(s-u)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(u\right)\left(\widehat{u}\left(u\right)-x\left(u\right)\right)|]duds\hfill \\ \hfill {\displaystyle }& & +\sum _{j=1}^q|{\zeta }_j|{\int }_a^{{\sigma }_j}[{\displaystyle \frac{{({\sigma }_j-s)}^{{\rho }_1+{\omega }_1-1}}{\Gamma ({\rho }_1+{\omega }_1)}}|f(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-f(s,x\left(s\right),y\left(s\right))|\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{({\sigma }_j-s)}^{{\omega }_1-1}}{\Gamma \left({\omega }_1\right)}}|{\mu }_1\left(s\right)\left(\widehat{u}\left(s\right)-x\left(s\right)\right)|]ds\hfill \\ \hfill {\displaystyle }& & +{\int }_a^b[{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\rho }_2+{\omega }_2-1}}{\Gamma ({\rho }_2+{\omega }_2)}}{\displaystyle \frac{|g(s,\widehat{u}\left(s\right),\widehat{v}\left(s\right))-g(s,x\left(s\right),y\left(s\right))|}{s}}\hfill \\ \hfill {\displaystyle }& & +{\displaystyle \frac{{\left(\log \frac{b}{s}\right)}^{{\omega }_2-1}}{\Gamma \left({\omega }_2\right)}}{\displaystyle \frac{|{\mu }_2\left(s\right)\left(\widehat{v}\left(s\right)-y\left(s\right)\right)|}{s}}]ds\hfill \\ & \le & {\mathcal{U}}_1{\epsilon }_1+{\mathcal{U}}_2{\epsilon }_2+\left({\mathcal{K}}_1{\mathcal{U}}_1+{\mathcal{K}}_2{\mathcal{U}}_2\right)(\parallel \widehat{u}-x\parallel +\parallel \widehat{v}-y\parallel )+{\mathcal{U}}_3\parallel \widehat{u}-x\parallel +{\mathcal{U}}_4\parallel \widehat{v}-y\parallel .\hfill \end{array}$$

In a similar fashion, one can find that

$$\begin{array}{ccc}\hfill \parallel \widehat{v}-y\parallel & \le & {\mathcal{V}}_1{\epsilon }_1+{\mathcal{V}}_2{\epsilon }_2+\left({\mathcal{K}}_1{\mathcal{V}}_1+{\mathcal{K}}_2{\mathcal{V}}_2\right)(\parallel \widehat{u}-x\parallel +\parallel \widehat{v}-y\parallel )+{\mathcal{V}}_3\parallel \widehat{u}-x\parallel +{\mathcal{V}}_4\parallel \widehat{v}-y\parallel .\hfill \end{array}$$

Thus, by condition (24), we get

$$\begin{array}{ccc}\hfill \parallel (\widehat{u},\widehat{v})-(x,y)\parallel & \le & \parallel \widehat{u}-x\parallel +\parallel \widehat{v}-y\parallel \hfill \\ & \le & ({\mathcal{U}}_1+{\mathcal{V}}_1){\epsilon }_1+({\mathcal{U}}_2+{\mathcal{V}}_2){\epsilon }_2\hfill \\ & & +\left({\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\right)\left(\parallel \widehat{u}-x\parallel +\parallel \widehat{v}-y\parallel \right)\hfill \\ & & +\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)(\parallel \widehat{u}-x\parallel +\parallel \widehat{v}-y\parallel ).\hfill \end{array}$$

Therefore,

$$\parallel \le (\widehat{u},\widehat{v})-(x,y)\parallel \le {\displaystyle \frac{({\mathcal{U}}_1+{\mathcal{V}}_1){\epsilon }_1+({\mathcal{U}}_2+{\mathcal{V}}_2){\epsilon }_2}{1-\left[{\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{U}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\right]-\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)}}.$$

(32)

Letting ${\displaystyle c=(c_f,c_g)=\left(\frac{{\mathcal{U}}_1+{\mathcal{V}}_1}{1-\mathbb{H}},\frac{{\mathcal{U}}_2+{\mathcal{V}}_2}{1-\mathbb{H}}\right)}$, we get

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

where $\mathbb{H}=\left[{\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)\right]+\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)<1$, by the condition (24). Hence, system (1) and (2) is Ulam–Hyers–stable. Furthermore, it is generalized Ulam–Hyers stable since $\parallel (\widehat{u},\widehat{v})-(x,y)\parallel \le \Lambda \left(\epsilon \right)$ with $\Lambda \left(\epsilon \right)=c{\epsilon }^T,\Lambda (0,0)=0$. □

5. Examples

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

Example 1. 

Consider a coupled system of nonlinear Langevin equations

$$\left\{\begin{array}{c}{\displaystyle {}^{RL}D^{\frac{7}{10}}\left({}^{RL}D^{\frac{37}{10}}+\frac{{\tau }^2}{100}\right)x\left(\tau \right)=f(\tau ,x\left(\tau \right),y\left(\tau \right)),\tau \in [1,3],}\hfill \\ {\displaystyle {}^{H}D^{\frac{8}{10}}\left({}^{H}D^{\frac{27}{10}}+\frac{1}{\tau }\right)y\left(\tau \right)=g(\tau ,x\left(\tau \right),y\left(\tau \right)),\tau \in [1,3],}\hfill \end{array}\right.$$

(33)

subject to the boundary data

$$\left\{\begin{array}{c}{\displaystyle x\left(1\right)=0,{}^{RL}D^{27/10}x\left(1\right)=0,{}^{RL}D^{17/10}x\left(1\right)=0,{}^{RL}D^{7/10}x\left(1\right)=0,}\hfill \\ {\displaystyle y\left(1\right)=0,{}^{H}D^{17/10}y\left(1\right)=0,{}^{H}D^{7/10}y\left(1\right)=0,}\hfill \\ {\displaystyle x\left(3\right)=15{\int }_1^3\frac{1}{s}y\left(s\right)ds+\sum _{i=1}^3{\eta }_{i}y\left({\alpha }_i\right),}\hfill \\ {\displaystyle y\left(3\right)=10{\int }_1^{3}x\left(s\right)ds+\sum _{j=1}^4{\zeta }_{j}x\left({\sigma }_j\right),}\hfill \end{array}\right.$$

(34)

with ${\rho }_1=0.7, {\rho }_2=0.8, {\omega }_1=3.7, {\omega }_2=2.7, n=4, m=3, {\lambda }_1=15, {\lambda }_2=10, {\alpha }_1=1.25, {\alpha }_2=1.5, {\alpha }_3=1.75, {\sigma }_1=2, {\sigma }_2=2.25, {\sigma }_3=2.5, {\sigma }_4=2.75, {\zeta }_1=1, {\zeta }_2=3, {\zeta }_3=2, {\zeta }_4=7, {\eta }_1=1/9, {\eta }_2=1/7, {\eta }_3=1/3, {\mu }_1\left(\tau \right)={\displaystyle \frac{{\tau }^2}{100}}, {\mu }_2\left(\tau \right)={\displaystyle \frac{1}{\tau }}$.

Using the given values, we find that ${\mathcal{U}}_1\approx 0.88005111,{\mathcal{U}}_2\approx 0.02069756,{\mathcal{U}}_3\approx 0.03393529$, ${\mathcal{U}}_4\approx 0.04596575,{\mathcal{V}}_1\approx 0.18395889,{\mathcal{V}}_2\approx 0.21617910$, ${\mathcal{V}}_3\approx 0.00709356$, and ${\mathcal{V}}_4\approx 0.48009696$ (${\mathcal{U}}_p$ and ${\mathcal{V}}_p$, $p=1,2,3,4$, are given in (17)).

(a)

To illustrate Theorem 3, for $\tau \in [1,3]$, we take

$$\begin{array}{ccc}\hfill & & {\displaystyle f(\tau ,x\left(\tau \right),y\left(\tau \right))=\frac{1}{10}+\frac{\cos x\left(\tau \right)}{25{\tau }^2+15}+\frac{|y(\tau )|}{2{(\tau +4)}^2(1+|y\left(\tau \right)|)},}\hfill \\ & & {\displaystyle g(\tau ,x\left(\tau \right),y\left(\tau \right))=\frac{1}{\sqrt{8{\tau }^2+1}}+\frac{x\left(\tau \right)\sin y\left(\tau \right)}{7{\tau }^3+8}+\frac{|y\left(\tau \right)|}{{(\tau +4)}^2\left(\sqrt{1+{|x\left(\tau \right)|}^2}\right)},}\hfill \end{array}$$

(35)

and note that

$$|f(\tau ,x,y)|\le {\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{40}}|x|+{\displaystyle \frac{1}{50}}|y|,$$

$$|g(\tau ,x,y)|\le {\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{15}}|x|+{\displaystyle \frac{1}{25}}|y|;$$

that is, the assumption $\left(H_1\right)$ holds with ${\gamma }_0={\displaystyle \frac{1}{10}},{\gamma }_1={\displaystyle \frac{1}{40}},{\gamma }_2={\displaystyle \frac{1}{50}},{\delta }_0={\displaystyle \frac{1}{3}},{\delta }_1={\displaystyle \frac{1}{15}},{\delta }_2={\displaystyle \frac{1}{25}}$.

Moreover, we find that ${\mathcal{F}}_1\approx 0.08342089<1$ and ${\mathcal{F}}_2\approx 0.55681798<1$. Thus, the hypotheses of Theorem 3 are verified. Therefore, the conclusion of Theorem 3 applies to problem (33) and (34) with f and g given in (35).

(b)

For illustrating Theorem 4, we take

$$\begin{array}{c}\hfill f(\tau ,x\left(\tau \right),y\left(\tau \right))={\displaystyle \frac{1}{6(\tau +9)}}{tan}^{-1}x\left(\tau \right)+{\displaystyle \frac{\cos y\left(\tau \right)}{12\sqrt{(\tau +24)}}}+15e^{\tau },\tau \in [1,3],\\ \hfill g(\tau ,x\left(\tau \right),y\left(\tau \right))={\displaystyle \frac{1}{5\sqrt{30{\tau }^2+70}}}\left({\displaystyle \frac{|x(\tau )|}{1+|x(\tau )|}}+\sin y\left(\tau \right)\right)+25{\tau }^2,\tau \in [1,3].\end{array}$$

(36)

For each $\tau \in [1,3]$ and $x_1,y_1,x_2,y_2\in \mathbb{R}$, we notice that

$$|f(\tau ,x_2,y_2)-f(\tau ,x_1,y_1)|\le {\displaystyle \frac{1}{60}}(|x_2-x_1|+|y_2-y_1|),$$

$$|g(\tau ,x_2,y_2)-g(\tau ,x_1,y_1)|\le {\displaystyle \frac{1}{50}}(|x_2-x_1|+|y_2-y_1|).$$

Thus, the condition ($H_2$) holds true with ${\mathcal{K}}_1={\displaystyle \frac{1}{60}}$ and ${\mathcal{K}}_2={\displaystyle \frac{1}{50}}$. Further, we have

$$\begin{array}{c}\hfill {\displaystyle {\mathcal{K}}_1({\mathcal{U}}_1+{\mathcal{V}}_1)+{\mathcal{K}}_2({\mathcal{U}}_2+{\mathcal{V}}_2)+\max ({\mathcal{U}}_3+{\mathcal{V}}_3,{\mathcal{U}}_4+{\mathcal{V}}_4)\approx 0.54853375<1.}\end{array}$$

Clearly, the hypothesis of Theorem 4 is satisfied. Hence, problems (33) and (34) with the nonlinear functions f and g given by (36) have a unique solution on $\mathcal{T}$. The numerical solutions for problem (33) and (34) with the nonlinearities f and g given in (36) for different values of ${\rho }_1$, ${\rho }_2$, ${\omega }_1$ and ${\omega }_2$ are shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6.

Example 2. 

Problem (33) and (34) with f and g given in (36) is Ulam–Hyers-stable as well as generalized Ulam–Hyers-stable, since $\mathbb{H}\approx 0.54853375<1$. The Ulam–Hyers stability for problem (33) and (34) with the nonlinearities f and g given in (36) and ${\varphi }_1\left(\tau \right)={\displaystyle \frac{1}{\tau +500}}$, ${\varphi }_2\left(\tau \right)={\displaystyle \frac{1}{\tau +1000}}$ for different values of ${\rho }_1,{\rho }_2$ and ${\omega }_1,{\omega }_2$ is depicted in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

6. Discussion

We used the fixed–point operator equation $(x,y)=\mathcal{W}(x,y)$, where $\mathcal{W}$ is defined in (14), to compute the numerical approximation of the solution $(x,y)$ for problem (33) and (34) with the nonlinearities f and g given in (36) (a similar argument holds for the solution $(\widehat{u},\widehat{v})$ for (31)). We applied a composite Simpson’s rule using the MATLAB R2024a (24.1.0.2537033, MathWorks, Inc., Galway, Ireland) platform to obtain the numerical approximation of solutions.

The value for each numerical solution depends on the condition (24), which changes by varying the values of fractional parameters ${\rho }_j,{\omega }_j,j=1,2$. We denoted the left-hand side of the condition (24) by $\mathbb{H}$ and calculated its value for each case and placed it below each numerical simulation in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 to demonstrate that the condition (24) is satisfied, i.e., each value is less than 1. Looking at the profile of solutions $(x,y)$ in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, the gap between the graphical solutions x and y decreases with increasing values of fractional parameters ${\rho }_j,{\omega }_j,j=1,2$.

The Ulam–Hyers stability for the solutions $(\widehat{u},\widehat{v})$, associated with the numerical solutions $(x,y)$ presented in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, is demonstrated in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. In the same pattern, we can compute the numerical approximation for the solutions $(x,y)$ for different combinations of values of fractional parameters ${\rho }_j,{\omega }_j,j=1,2$.

7. Conclusions

We developed the existence and Ulam–Hyers stability criteria for solutions of a nonlocal nonlinear Langevin–type coupled integral boundary value problem involving Riemann–Liouville and Hadamard fractional derivative operators. The results accomplished in this paper are novel in the given setting and specialize to some new results. For example, by taking ${\lambda }_1={\lambda }_2=0$, our results correspond to the ones for the nonlocal coupled multipoint boundary conditions:

$$\begin{array}{c}{\displaystyle x\left(a\right)=0,{}^{RL}D^{{\omega }_1-v_1}x\left(a\right)=0,v_1=1,2,3,\dots ,n-1,}\hfill \\ {\displaystyle y\left(a\right)=0,{}^{H}D^{{\omega }_2-v_2}y\left(a\right)=0,v_2=1,2,3,\dots ,m-1,}\hfill \\ {\displaystyle x\left(b\right)=\sum _{i=1}^p{\eta }_{i}y\left({\alpha }_i\right),y\left(b\right)=\sum _{j=1}^q{\zeta }_{j}x\left({\sigma }_j\right),{\alpha }_i,{\sigma }_j,\in (a,b),{\eta }_i,{\zeta }_j\in \mathbb{R}.}\hfill \end{array}$$

If we choose ${\eta }_i={\zeta }_j=0,i=1,\dots ,p,j=1,\dots ,q$, then we get the results associated with coupled integral boundary conditions:

$$\begin{array}{c}{\displaystyle x\left(a\right)=0,{}^{RL}D^{{\omega }_1-v_1}x\left(a\right)=0,v_1=1,2,3,\dots ,n-1,}\hfill \\ {\displaystyle y\left(a\right)=0,{}^{H}D^{{\omega }_2-v_2}y\left(a\right)=0,v_2=1,2,3,\dots ,m-1,}\hfill \\ {\displaystyle x\left(b\right)={\lambda }_1{\int }_a^b\frac{1}{s}y\left(s\right)ds,y\left(b\right)={\lambda }_2{\int }_a^{b}x\left(s\right)ds,{\lambda }_1,{\lambda }_2\ge 0.}\hfill \end{array}$$

In future work, we plan to work on the multivalued variant of problem (1) and (2).