Existence and Ulam–Hyers Stability Results for a System of Coupled Generalized Liouville–Caputo Fractional Langevin Equations with Multipoint Boundary Conditions

Abstract

We study the existence and uniqueness of solutions for coupled Langevin differential equations of fractional order with multipoint boundary conditions involving generalized Liouville–Caputo fractional derivatives. Furthermore, we discuss Ulam–Hyers stability in the context of the problem at hand. The results are shown with examples. Results are asymmetric when a generalized Liouville–Caputo fractional derivative $\left(\rho \right)$ parameter is changed.

1. Introduction

The study of operators of fractional order, including integral and derivative, over either real or complex domains is the focus of fractional calculus. Due to its active development and applications in the fields of physics, mechanics, chemistry, engineering, etc., fractional differential equations (FDEs) have become a popular topic of study among scholars. The books [1,2,3,4,5,6,7] are recommended for readers interested in the systematic development of the topic. In mathematical physics, the Langevin equation (LE) is a potent tool for modelling anomalous diffusion and other phenomena in a systematic way. Harmonic oscillators, price index variations [8], and other related processes are examples of this type of process. The theory of critical dynamics also extensively makes use of a general Langevin equation for noise sources with correlations [9]. To comprehend the nature of the quantum noise, a generalized Langevin equation (GLE) [10] can be employed. The LE has a highly elegant and rich role in fractional systems, such as fractional reaction–diffusion systems [11,12]. It is suggested that the fractional analogue of the standard LE, also known as the stochastic differential equation, be used in cases where the separation between microscopic and macroscopic time scales is not obvious; see, for instance, [13]. For a GLE of fractional order of the Liouville–Caputo fractional derivative, the author of [14] investigated moments, variances, location, and velocity correlation. Comparisons were made between the outcomes and those obtained for the same GLE. The aforementioned papers [15,16,17,18,19,20,21] and their relevant references contain recent studies on the LE with varied boundary conditions. The study of fractional calculus has become a significant area of study because of the multiple applications it has in the technical sciences, social sciences, and engineering professions. Differential and integral operators based on a fractional order are thought to be more realistic and useful than their integer-order counterparts because they can show the history of ongoing phenomena and processes. Recently, the literature on the subject has included Hadamard, Caputo(Liouville–Caputo), Riemann–Liouville type derivatives, among others, as well as FDEs. See citations [22,23,24,25,26,27,28,29,30], as well as the references, for a few recent publications on the topic. The authors in [31] discussed the existence of solutions for Langevin FDEs using Liouville–Caputo derivatives:

$$\left\{\begin{array}{c}{}_c^{\rho }{\mathcal{D}}_{a^{+}}^{\alpha }{(}_c^{\rho }{\mathcal{D}}_{a^{+}}^{\beta }+\lambda )x\left(t\right)=f(t,x\left(t\right)),t\in \mathcal{J}:=[a,T],\hfill \\ \lambda \in \mathbb{R},1<\alpha \le 2,0<\beta <1,\rho >0\hfill \end{array}\right.$$

(1)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(a\right)=0,x\left(\eta \right)=0,x\left(T\right)={\mu }^{\rho }{\mathcal{I}}_{a^{+}}^{\gamma }x\left(\xi \right),\gamma >0,\rho >0,\hfill \end{array}\right.\end{array}$$

(2)

where ${}_c^{\rho }{\mathcal{D}}_{a^{+}}^{\alpha }$, ${}_c^{\rho }{\mathcal{D}}_{a^{+}}^{\beta }$ denote the generalized Liouville–Caputo fractional derivatives (GLCFD), ${}^{\rho }{\mathcal{I}}_{a^{+}}^{\gamma }$ is the generalized fractional integral (GFI). The main results were proven using a fixed-point index theory. Applications to differential equations in science inevitably lead to multipoint boundary value problems. A dynamical system with m degrees of freedom, for instance, might have exactly m examples that are seen at m different times. An m-point boundary value problem is a mathematical representation of such problems. Multipoint problems for differential equations are a subset of interface problems and are therefore amenable to a variety of solutions. The study of fractional differential equations with multipoint boundary conditions has attracted the interest of numerous researchers (see [32,33,34,35] and the references given therein). We demonstrate the existence, uniqueness of solutions and Ulam–Hyers stability for the following generalized Langevin fractional differential system with multipoint boundary conditions, which was inspired by previous research, by utilizing the fixed-point theorems:

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_1}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right)=f(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,\mathcal{S}],\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_2}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right)=g(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,\mathcal{S}],\hfill \end{array}\right.$$

(3)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=0,y\left(0\right)=0,x\left(\mathcal{S}\right)=ϵ{\sum }_{j=1}^k{\varsigma }_{j}y\left({\varpi }_j\right),y\left(\mathcal{S}\right)=\pi {\sum }_{j=1}^k{\varrho }_{j}x\left({\sigma }_j\right),\hfill \\ 0<{\sigma }_1<{\varpi }_1<\cdots <{\sigma }_k<{\varpi }_k<\mathcal{S},\hfill \end{array}\right.\end{array}$$

(4)

where ${}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_1},{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_2}$, ${}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1},{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}$ are the GLCFD of order $1<{\xi }_1,{\xi }_2\le 2$, $0<{\zeta }_1,{\zeta }_2<1$, and $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $ϵ,\pi \in \mathbb{R}$. The requirement states that the unknown function’s value at the right endpoint of the specified interval, $\iota =\mathcal{S}$, must be proportional to its values on different multipoint values of unknown functions with ${\sigma }_j,{\varpi }_j,j=1,2,\cdots k$, where ${\varsigma }_j,{\varrho }_j,j=1,2,\cdots k$ are arbitrary constants. The GLCFD is converted into the differentially effective Caputo sense when $\rho =1$. It should be noticed that the multipoint strip boundary condition in (4) is new in the context of generalized Liouville–Caputo fractional differential equations and can be understood as the value of a known function at $\mathcal{S}$ being proportional to the discrete values of the unknown function at ${\varpi }_i,{\sigma }_i$, $i=1,2,\dots ,k$. By employing the fixed-point technique, we investigate the existence and uniqueness solutions of nonlocal generalized Liouville–Caputo fractional boundary value problem with discrete boundary conditions (3) and (4). Differently from the above mentioned works, we obtain the Ulam–Hyers stability solutions of the nonlocal boundary value problem of the generalized Liouville–Caputo type fractional differential Equations (3) and (4) by using conventional functional analysis. To the best of our knowledge, boundary value problems’ (BVPs) stability analysis is still in its early development. The fundamental contribution of this research is to investigate the existence of Ulam–Hyers stability solutions. In addition, we show the problems (3) and (4) used by the fixed-point theorems of Leray–Schauder and Banach to demonstrate the existence and uniqueness of the solutions. The remainder of the paper is structured as follows: In Section 2, we review some fundamental ideas of fractional calculus and find the integral solutions to the given problems’ linear versions. The existence results for problems (3) and (4) derived by using Leray–Schauder’s nonlinear alternative and Banach’s contraction mapping principle are presented in Section 3. Section 4 looks at the Ulam–Hyers stability of the provided systems (3) and (4) under particular circumstances. Examples are provided in Section 6 to further clarify the study’s findings.

2. Preliminaries

This section begins with the fundamental definitions of fractional calculus. Later, we demonstrate an auxiliary lemma that is crucial in formulating a fixed-point problem related to the topic at hand. We define space $\mathcal{P}=\left\{x\right(\iota ):x(\iota )\in \mathcal{C}(\mathcal{E},\mathbb{R}\left)\right\}$ equipped with the norm $\left|\right|x\left|\right|=sup\left\{\right|x\left(\iota \right)|,\iota \in \mathcal{E}\}$ as a Banach space. Moreover, $\mathcal{Q}=\left\{y\right(\iota ):y(\iota )\in \mathcal{C}(\mathcal{E},\mathbb{R}\left)\right\}$ equipped with the norm $\left|\right|y\left|\right|=sup\left\{\right|y\left(\iota \right)|,\iota \in \mathcal{E}\}$ is a Banach space. Then, the product space $(\mathcal{P}\times \mathcal{Q},|\left|\right(x,y\left)\right|\left|\right)$ is also a Banach space with norm $\left|\right|(x,y)\left|\right|=\left|\right|x\left|\right|+\left|\right|y\left|\right|$.

 Definition 1

([36]). The left- and right-sided GFIs of $f\in {\mathcal{Z}}_b^y(c,d)$ of order $\xi >0$ and $\rho >0$, for $-\infty <c<\iota <d<\infty$, are defined as follows:

$$\begin{array}{c}\hfill {(}^{\rho }{\mathcal{I}}_{c^{+}}^{\xi }f)\left(\iota \right)=\frac{{\rho }^{1-\xi }}{\mathsf{\Gamma }\left(\xi \right)}{\int }_c^{\iota }\frac{{\theta }^{\rho -1}}{{({\iota }^{\rho }-{\theta }^{\rho })}^{1-\xi }}f\left(\theta \right)d\theta ,\end{array}$$

(5)

$$\begin{array}{c}\hfill {(}^{\rho }{\mathcal{I}}_{d^{-}}^{\zeta }f)\left(\iota \right)=\frac{{\rho }^{1-\xi }}{\mathsf{\Gamma }\left(\xi \right)}{\int }_{\iota }^d\frac{{\theta }^{\rho -1}}{{({\theta }^{\rho }-{\iota }^{\rho })}^{1-\xi }}f\left(\theta \right)d\theta .\end{array}$$

(6)

 Definition 2

([37]). The generalized fractional derivatives (GFDs) which are associated with GFIs (5) and (6), for $0\le c<\iota <d<\infty$, are defined as follows:

$$\begin{array}{cc}\hfill {(}^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }f)\left(\iota \right)& ={\left({\iota }^{1-\rho }\frac{d}{d\iota }\right)}^n{(}^{\rho }{\mathcal{I}}_{c^{+}}^{n-\zeta }f)\left(\iota \right)\hfill \\ & =\frac{{\rho }^{\zeta -n+1}}{\mathsf{\Gamma }(n-\zeta )}{\left({\iota }^{1-\rho }\frac{d}{d\iota }\right)}^n{\int }_c^{\iota }\frac{{\theta }^{\rho -1}}{{({\iota }^{\rho }-{\theta }^{\rho })}^{\zeta -n+1}}f\left(\theta \right)d\theta ,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {(}^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f)\left(\iota \right)& ={\left(-{\iota }^{1-\rho }\frac{d}{d\iota }\right)}^n{(}^{\rho }{\mathcal{I}}_{d^{-}}^{n-\zeta }f)\left(\iota \right)\hfill \\ & =\frac{{\rho }^{\zeta -n+1}}{\mathsf{\Gamma }(n-\zeta )}{\left(-{\iota }^{1-\rho }\frac{d}{d\iota }\right)}^n{\int }_{\iota }^d\frac{{\theta }^{\rho -1}}{{({\iota }^{\rho }-{\theta }^{\rho })}^{\zeta -n+1}}f\left(\theta \right)d\theta ,\hfill \end{array}$$

(8)

if the integrals exist.

 Definition 3

([38]). The above GFDs define the left- and right-sided generalized Liouville–Caputo type fractional derivatives of $f\in \mathcal{A}{\mathcal{C}}_{\gamma }^n[c,d]$ of order $\zeta \ge 0$

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }f\left(z\right){=}^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }\left[f\left(\iota \right)-\sum _{k=0}^{n-1}\frac{{\gamma }^{k}f\left(c\right)}{k!}{\left(\frac{{\iota }^{\rho }-c^{\rho }}{\rho }\right)}^k\right]\left(z\right),\gamma =z^{1-\rho }\frac{d}{dz},\end{array}$$

(9)

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f\left(z\right){=}^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }\left[f\left(\iota \right)-\sum _{k=0}^{n-1}\frac{{(-1)}^k{\gamma }^{k}f\left(d\right)}{k!}{\left(\frac{d^{\rho }-{\iota }^{\rho }}{\rho }\right)}^k\right]\left(z\right),\gamma =z^{1-\rho }\frac{d}{dz},\end{array}$$

(10)

when $n=\left[\zeta \right]+1.$

 Lemma 1

([38]). 1. If $\zeta \notin \mathbb{N}$,

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }f\left(\iota \right)=\frac{1}{\mathsf{\Gamma }(n-\zeta )}{\int }_c^{\iota }{\left(\frac{{\iota }^{\rho }-{\theta }^{\rho }}{\rho }\right)}^{n-\zeta -1}\frac{\left({\gamma }^{n}f\right)\left(\theta \right)d\theta }{{\theta }^{1-\rho }}{=}^{\rho }{\mathcal{I}}_{c^{+}}^{n-\zeta }\left({\gamma }^{n}f\right)\left(\iota \right),\end{array}$$

(11)

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f\left(\iota \right)=\frac{1}{\mathsf{\Gamma }(n-\zeta )}{\int }_{\iota }^d{\left(\frac{{\theta }^{\rho }-{\iota }^{\rho }}{\rho }\right)}^{n-\zeta -1}\frac{{(-1)}^n\left({\gamma }^{n}f\right)\left(\theta \right)d\theta }{{\theta }^{1-\rho }}{=}^{\rho }{\mathcal{I}}_{d^{-}}^{n-\zeta }\left({\gamma }^{n}f\right)\left(\iota \right).\end{array}$$

(12)

2. If $\zeta \in \mathbb{N}$,

$$\begin{array}{c}\hfill {}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }f={\gamma }^{n}f,{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f={(-1)}^n{\gamma }^{n}f.\end{array}$$

(13)

 Lemma 2

([38]). Let $f\in \mathcal{A}{\mathcal{C}}_{\gamma }^n[c,d]$ or ${\mathcal{C}}_{\gamma }^n[c,d]$ and $\zeta \in \mathbb{R}$. Then,

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{c^{+}}^{\zeta }{}_C^{\rho }{\mathcal{D}}_{c^{+}}^{\zeta }f\left(\iota \right)=f\left(\iota \right)-\sum _{k=0}^{n-1}\frac{{\gamma }^{k}f\left(c\right)}{k!}{\left(\frac{{\iota }^{\rho }-c^{\rho }}{\rho }\right)}^k,\end{array}$$

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{d^{-}}^{\zeta }{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f\left(\iota \right)=f\left(\iota \right)-\sum _{k=0}^{n-1}\frac{{(-1)}^k{\gamma }^{k}f\left(d\right)}{k!}{\left(\frac{d^{\rho }-{\iota }^{\rho }}{\rho }\right)}^k.\end{array}$$

In particular, for $0<\zeta \le 1$, we have

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{c^{+}}^{\zeta }{}_C^{\rho }{\mathcal{D}}_{c+}^{\zeta }f\left(\iota \right)=f\left(\iota \right)-f\left(c\right),{}^{\rho }{\mathcal{I}}_{d^{-}}^{\zeta }{}_C^{\rho }{\mathcal{D}}_{d^{-}}^{\zeta }f\left(\iota \right)=f\left(\iota \right)-f\left(d\right).\end{array}$$

In order to facilitate the computation, we present the following notations:

$$\begin{array}{cc}\hfill & \widehat{{\mathcal{E}}_1}=\frac{{\mathcal{S}}^{\rho {\zeta }_1}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)},\widehat{{\mathcal{E}}_2}=\frac{\pi {\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)},\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill {\mathcal{E}}_1=\frac{{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)},{\mathcal{E}}_2=\frac{ϵ{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)},\end{array}$$

(15)

$$\mathcal{G}=\widehat{{\mathcal{E}}_1}{\mathcal{E}}_2-{\mathcal{E}}_1\widehat{{\mathcal{E}}_2}\ne 0,$$

(16)

$${\delta }_1\left(\iota \right)=\left(\frac{{\iota }^{\rho {\zeta }_1}}{\rho {\zeta }_1\mathsf{\Gamma }\left({\zeta }_1+1\right)\mathcal{G}}\right),{\delta }_2\left(\iota \right)=\left(\frac{{\iota }^{\rho {\zeta }_2}}{\rho {\zeta }_2\mathsf{\Gamma }\left({\zeta }_2+1\right)\mathcal{G}}\right).$$

(17)

 Lemma 3.

Let $\widehat{f},\widehat{g}\in C(0,\mathcal{S})\cup \mathcal{L}(0,\mathcal{S}),x,y\in \mathcal{A}{\mathcal{C}}_{\gamma }^2\left(\mathcal{E}\right)$, and $\mathsf{\Lambda }\ne 0.$ The solution of the system of coupled Langevin fractional BVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_1}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right)=\widehat{f}\left(\iota \right),\iota \in \mathcal{E}:=[0,\mathcal{S}],\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_2}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right)=\widehat{g}\left(\iota \right),\iota \in \mathcal{E}:=[0,\mathcal{S}],\hfill \\ x\left(0\right)=0,y\left(0\right)=0,x\left(\mathcal{S}\right)=ϵ{\sum }_{j=1}^k{\varsigma }_{j}y\left({\varpi }_j\right),y\left(\mathcal{S}\right)=\pi {\sum }_{j=1}^k{\varrho }_{j}x\left({\sigma }_j\right),\hfill \\ 0<{\sigma }_1<{\varpi }_1<\cdots <{\sigma }_k<{\varpi }_k<\mathcal{S},\hfill \end{array}\right.\end{array}$$

(18)

is given by

$$\begin{array}{cc}\hfill x\left(\iota \right)& {=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\iota \right)-{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\iota \right)\hfill \\ & +{\delta }_1\left(\iota \right)[{\mathcal{E}}_2\left(ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right)\right)\hfill \\ & +{\mathcal{E}}_1\left(\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\right)]\hfill \end{array}$$

(19)

and

$$\begin{array}{cc}\hfill y\left(\iota \right)& {=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\iota \right)-{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\iota \right)\hfill \\ & +{\delta }_2\left(\iota \right)[\widehat{{\mathcal{E}}_2}\left(ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right)\right)\hfill \\ & +\widehat{{\mathcal{E}}_1}\left(\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\right)].\hfill \end{array}$$

(20)

 Proof. 

Applying operators ${}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1}{,}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2}$ to (18) and using Lemma 2, we get

$$\begin{array}{c}\hfill ({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right){=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1}\widehat{f}\left(\iota \right)+a_1,\end{array}$$

(21)

$$\begin{array}{c}\hfill ({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right){=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2}\widehat{g}\left(\iota \right)+b_1,\end{array}$$

(22)

respectively, for some $a_1,b_1\in \mathcal{R}$. When ${}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}{,}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}$ are applied to the FDEs in (21) and (22), the solution of the Langevin FDEs in (18) for $\iota \in \mathcal{E}$ is

$$\begin{array}{c}\hfill x\left(\iota \right){=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\iota \right)-{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\iota \right)+a_1\frac{{\iota }^{\rho {\zeta }_1}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+a_2,\end{array}$$

(23)

$$\begin{array}{c}\hfill y\left(\iota \right){=}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\iota \right)-{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\iota \right)+b_1\frac{{\iota }^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+b_2,\end{array}$$

(24)

respectively, for some $a_2,b_2\in \mathcal{R}$. By utilizing the conditions $x\left(0\right)=y\left(0\right)=0$ in (23) and (24), respectively, we get $a_2=b_2=0$. Then, using the multipoint boundary conditions, we get:

$$\begin{array}{c}\hfill \sum _{j=1}^k{\varrho }_{j}x\left({\sigma }_j\right)=\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-{\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right)+a_1\sum _{j=1}^k\frac{{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)},\end{array}$$

(25)

$$\begin{array}{c}\hfill \sum _{j=1}^k{\varsigma }_{j}y\left({\varpi }_j\right)=\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right)+b_1\sum _{j=1}^k\frac{{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)},\end{array}$$

(26)

which, when combined with the boundary conditions $x\left(\mathcal{S}\right)=ϵ{\sum }_{j=1}^{k}y\left({\varpi }_j\right),y\left(\mathcal{S}\right)=\pi {\sum }_{j=1}^{k}x\left({\sigma }_j\right)$, gives the following results:

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)-{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right)+a_1\frac{{\mathcal{S}}^{\rho {\zeta }_1}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}=ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right)\\ \hfill +b_1ϵ\sum _{j=1}^k\frac{{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)},\end{array}$$

(27)

$$\begin{array}{c}\hfill {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)-{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)+b_1\frac{{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}=\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right)\\ \hfill +a_1\pi \sum _{j=1}^k\frac{{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}.\end{array}$$

(28)

Next, we obtain

$$\begin{array}{c}\hfill a_1\widehat{{\mathcal{E}}_1}-b_1{\mathcal{E}}_1=ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right),\end{array}$$

(29)

$$\begin{array}{c}\hfill b_1{\mathcal{E}}_2-a_2\widehat{{\mathcal{E}}_2}=\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right),\end{array}$$

(30)

using the notations (15) in (27) and (28), respectively. When the system of equations is solved, we find that (29) and (30) for $a_1$ and $b_1$ are

$$\begin{array}{ll}{a}_1=& \frac{1}{\mathcal{G}}[{\mathcal{E}}_2\left(ϵ{\displaystyle \sum _{j=1}^k}{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2{\displaystyle \sum _{j=1}^k}{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right)\right)\hfill \\ & +{\mathcal{E}}_1\left(\pi {\displaystyle \sum _{j=1}^k}{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1{\displaystyle \sum _{j=1}^k}{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\right)],\end{array}$$

(31)

$$\begin{array}{ll}{b}_1=& \frac{1}{\mathcal{G}}[\widehat{{\mathcal{E}}_2}\left(ϵ{\displaystyle \sum _{j=1}^k}{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left({\varpi }_j\right)-ϵ{\varphi }_2{\displaystyle \sum _{j=1}^k}{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right)\right)\\ & +\widehat{{\mathcal{E}}_1}\left(\pi {\displaystyle \sum _{j=1}^k}{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\widehat{f}\left({\sigma }_j\right)-\pi {\varphi }_1{\displaystyle \sum _{j=1}^k}{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\widehat{g}\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\right)].\end{array}$$

(32)

Substituting the values of $a_1,b_1$ in (23) and (24), respectively, we obtain the BVP solution (18). □

3. Main Results

We propose a fixed-point problem relevant to the problem in Lemma 3 as follows: $\mathsf{\Psi }:\mathcal{P}\times \mathcal{Q}\to \mathcal{P}\times \mathcal{Q}$ by

$$\begin{array}{c}\hfill \mathsf{\Psi }(x,y)\left(\iota \right)=({\mathsf{\Psi }}_1(x,y)\left(\iota \right),{\mathsf{\Psi }}_2(x,y)\left(\iota \right)),\end{array}$$

(33)

where

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_1(x,y)\left(\iota \right)=& {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}f(\iota ,x\left(\iota \right),y\left(\iota \right))-{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\iota \right)+{\delta }_1\left(\iota \right)\left[{\mathcal{E}}_2\right(ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}g({\varpi }_j,x\left({\varpi }_j\right),y\left({\varpi }_j\right))\hfill \\ & -ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right)-{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}f(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right))\hfill \\ & +{\mathcal{E}}_1(\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}f({\sigma }_j,x\left({\sigma }_j\right),y\left({\sigma }_j\right))-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right)\hfill \\ & -{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}g(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\left)\right],\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_2(x,y)\left(\iota \right)=& {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}g(\iota ,x\left(\iota \right),y\left(\iota \right))-{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\iota \right)+{\delta }_2\left(\iota \right)\left[\widehat{{\mathcal{E}}_2}\right(ϵ\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}g({\varpi }_j,x\left({\varpi }_j\right),y\left({\varpi }_j\right))\hfill \\ & -ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right){-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}f(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right))\hfill \\ & +\widehat{{\mathcal{E}}_1}(\pi \sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}f({\sigma }_j,x\left({\sigma }_j\right),y\left({\sigma }_j\right))-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right)\hfill \\ & {-}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}g(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\left)\right].\hfill \end{array}$$

(35)

For brevity, we use these notations:

$$\begin{array}{c}\hfill {\mathcal{U}}_1=\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)},\end{array}$$

(36)

$$\begin{array}{c}\hfill {\mathcal{V}}_1=\left|{\delta }_1\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right),\end{array}$$

(37)

$$\begin{array}{c}\hfill \widehat{{\mathcal{U}}_1}=\left|{\varphi }_1\right|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right),\end{array}$$

(38)

$$\begin{array}{c}\hfill \widehat{{\mathcal{V}}_1}=|{\delta }_1\left|\right|{\varphi }_2|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right),\end{array}$$

(39)

$$\begin{array}{c}\hfill {\mathcal{U}}_2=\left|{\delta }_2\right|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right),\end{array}$$

(40)

$$\begin{array}{c}\hfill {\mathcal{V}}_2=\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)},\end{array}$$

(41)

$$\begin{array}{c}\hfill \widehat{{\mathcal{U}}_2}=|{\delta }_2\left|\right|{\varphi }_1|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right),\end{array}$$

(42)

$$\begin{array}{c}\hfill \widehat{{\mathcal{V}}_2}=\left|{\varphi }_2\right|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right),\end{array}$$

(43)

$$\begin{array}{c}\hfill \mathsf{\Phi }=min\{1-[{\psi }_1({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_1}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2}],\\ \hfill 1-[{\psi }_2({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_2}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}]\}.\end{array}$$

(44)

Let $f,g:\mathcal{E}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be continuous functions.

• $\left({\mathcal{A}}_1\right)$ there exist constants ${\psi }_m,\widehat{{\psi }_m}\ge 0(m=1,2)$ and ${\psi }_0,\widehat{{\psi }_0}>0$ such that

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|f\right(\iota ,o_1,o_2\left)\right|& \le {\psi }_0+{\psi }_1|o_1|+{\psi }_2\left|o_2\right|,\hfill \\ \hfill \left|g\right(\iota ,o_1,o_2\left)\right|& \le \widehat{{\psi }_0}+\widehat{{\psi }_1}|o_1|+\widehat{{\psi }_2}\left|o_2\right|,\forall o_m\in \mathbb{R},m=1,2.\hfill \end{array}\end{array}$$
• (${\mathcal{A}}_2$) there exist constants ${\psi }_m,\widehat{{\psi }_m}\ge 0(m=1,2)$ such that

  $$\begin{array}{cc}\hfill |f(\iota ,o_1,o_2)-f(\iota ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & {\psi }_1|o_1-{\widehat{o}}_1|+{\psi }_2|o_2-{\widehat{o}}_2|,\hfill \\ \hfill |g(\iota ,o_1,o_2)-g(\iota ,{\widehat{o}}_1,{\widehat{o}}_2)|\le & \widehat{{\psi }_1}|o_1-{\widehat{o}}_1|+\widehat{{\psi }_2}|o_2-{\widehat{o}}_2|,\forall o_m,{\widehat{o}}_m\in \mathbb{R},m=1,2.\hfill \end{array}$$

 Theorem 1.

If the assumption $\left({\mathcal{A}}_1\right)$ is satisfied, then the problem in (3) and (4) has at least one solution on $\mathcal{E}$ if ${\psi }_1({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_1}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2}<1,{\psi }_2({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_2}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}<1$, where ${\mathcal{U}}_1,{\mathcal{V}}_1,\widehat{{\mathcal{U}}_1},\widehat{{\mathcal{V}}_1},{\mathcal{U}}_2,{\mathcal{V}}_2,\widehat{{\mathcal{U}}_2},\widehat{{\mathcal{V}}_2}$ are given by (36)–(43), respectively.

 Proof. 

In the first phase, we define operator $\mathsf{\Psi }:\mathcal{P}\times \mathcal{Q}\to \mathcal{P}\times \mathcal{Q}$ as being completely continuous. The operators ${\mathsf{\Psi }}_1$ and ${\mathsf{\Psi }}_2$ are continuous because the functions f and g are continuous. The operator $\mathsf{\Psi }$ is continuous as a result. For the purpose of illustrating how the uniformly bounded operator $\mathsf{\Psi }$ works, consider the bounded set $\mathsf{\Psi }\subset \mathcal{P}\times \mathcal{Q}$. Then, $\widehat{{\mathcal{N}}_1}$ and $\widehat{{\mathcal{N}}_2}$ are positive constants such that $\left|f(\iota ,x\left(\iota \right),y\left(\iota \right))\right|\le \widehat{{\mathcal{N}}_1},\left|g(\iota ,x\left(\iota \right),y\left(\iota \right))\right|\le \widehat{{\mathcal{N}}_2}$, $\forall (x,y)\in \mathsf{\Psi }$. Then, we have

$$\begin{array}{cc}\hfill |{\mathsf{\Psi }}_1(x,y)\left(\iota \right)|\le & {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}\left|f(\iota ,x\left(\iota \right),y\left(\iota \right))\right|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\iota \right)\right|\hfill \\ & +|{\delta }_1\left(\iota \right)\left|\right[|{\mathcal{E}}_2\left|\right(\left|ϵ\right|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g({\varpi }_j,x\left({\varpi }_j\right),y\left({\varpi }_j\right))\right|\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y\left({\varpi }_j\right)|+{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\mathcal{S}\right)\right|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f({\sigma }_j,x\left({\sigma }_j\right),y\left({\sigma }_j\right))|+|\pi \left|\right|{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left({\sigma }_j\right)\right|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))\right|+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}\left|y\left(\mathcal{S}\right)\right|\left)\right]\hfill \\ \hfill & \le {\widehat{\mathcal{N}}}_1\left\{\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right\}\hfill \\ \hfill & +\widehat{{\mathcal{N}}_2}\left\{|{\delta }_1|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\right\}\hfill \\ \hfill & +\left\{|{\varphi }_1|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\parallel x\parallel \hfill \\ \hfill & +\left\{|{\delta }_1||{\varphi }_2|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\parallel y\parallel ,\hfill \end{array}$$

when taking the norm and using (36)–(39), which yields for $(x,y)\in \mathsf{\Psi }$,

$$\begin{array}{c}\hfill \left|\right|{\mathsf{\Psi }}_1(x,y)\left|\right|\le {\mathcal{U}}_1{\widehat{\mathcal{N}}}_1+\widehat{{\mathcal{U}}_1}\parallel x\parallel +{\mathcal{V}}_1{\widehat{\mathcal{N}}}_2+\widehat{{\mathcal{V}}_1}\parallel y\parallel .\end{array}$$

(45)

Likewise, we obtain

$$\begin{array}{cc}\hfill \left|\right|{\mathsf{\Psi }}_2(x,y)\left|\right|\le & \widehat{{\mathcal{N}}_2}\left\{\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right\}\hfill \\ & +\widehat{{\mathcal{N}}_1}\left\{|{\delta }_2|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right)\right\}\hfill \\ & +\left\{|{\varphi }_2|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\parallel y\parallel \hfill \\ & +\left\{|{\delta }_2||{\varphi }_1|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\parallel x\parallel \hfill \\ \hfill \le & {\mathcal{U}}_2{\widehat{\mathcal{N}}}_1+\widehat{{\mathcal{U}}_2}\parallel x\parallel +{\mathcal{V}}_2{\widehat{\mathcal{N}}}_2+\widehat{{\mathcal{V}}_2}\parallel y\parallel ,\hfill \end{array}$$

(46)

using (40)–(43). We may infer that ${\mathsf{\Psi }}_1$ and ${\mathsf{\Psi }}_2$ are uniformly bounded based on the inequalities (45) and (46), which means that the operator $\mathsf{\Psi }$ is also uniformly bounded. Following that, we demonstrate that $\mathsf{\Psi }$ is equicontinuous. Let ${\iota }_1,{\iota }_2\in \mathcal{E}$ with ${\iota }_1<{\iota }_2$. Then, we have

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_1(x,y)\left({\iota }_2\right)-{\mathsf{\Psi }}_1(x,y)\left({\iota }_1\right)|\hfill \\ \hfill & {\le |}^{\rho }{\mathcal{I}}_{0+}^{{\xi }_1+{\zeta }_1}f({\iota }_2,x\left({\iota }_2\right),y\left({\iota }_2\right))-{}^{\rho }{\mathcal{I}}_{0+}^{{\xi }_1+{\zeta }_1}f({\iota }_1,x\left({\iota }_1\right),y\left({\iota }_1\right))|+|{\varphi }_1{\left|\right|}^{\rho }{\mathcal{I}}_{0+}^{{\zeta }_1}x\left({\iota }_2\right)-{}^{\rho }{\mathcal{I}}_{0+}^{{\zeta }_1}x\left({\iota }_1\right)|\hfill \\ & +|{\delta }_1\left({\iota }_2\right)-{\delta }_1\left({\iota }_1\right)|[|{\mathcal{E}}_2\left|\right(\left|ϵ\right|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g({\varpi }_j,x\left({\varpi }_j\right),y\left({\varpi }_j\right))\right|\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y\left({\varpi }_j\right)|+{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\mathcal{S}\right)\right|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f({\sigma }_j,x\left({\sigma }_j\right),y\left({\sigma }_j\right))|+|\pi \left|\right|{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left({\sigma }_j\right)\right|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))\right|+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}\left|y\left(\mathcal{S}\right)\right|\left)\right]\hfill \\ \hfill \le & \frac{{\rho }^{1-({\xi }_1+{\zeta }_1)}\widehat{{\mathcal{N}}_1}}{\mathsf{\Gamma }({\xi }_1+{\zeta }_1)}|{\int }_0^{{\iota }_1}\left[\frac{{\theta }^{\rho -1}}{{({\iota }_2^{\rho }-{\theta }^{\rho })}^{1-({\xi }_1+{\zeta }_1)}}-\frac{{\theta }^{\rho -1}}{{({\iota }_1^{\rho }-{\theta }^{\rho })}^{1-({\xi }_1+{\zeta }_1)}}\right]d\theta \hfill \\ & +{\int }_{{\iota }_1}^{{\iota }_2}\frac{{\theta }^{\rho -1}}{{({\iota }_2^{\rho }-{\theta }^{\rho })}^{1-\left({\zeta }_1\right)}}d\theta |\hfill \\ & \frac{{\rho }^{1-\left({\zeta }_1\right)}\parallel x\parallel }{\mathsf{\Gamma }\left({\zeta }_1\right)}|{\int }_0^{{\iota }_1}\left[\frac{{\theta }^{\rho -1}}{{({\iota }_2^{\rho }-{\theta }^{\rho })}^{1-\left({\zeta }_1\right)}}-\frac{{\theta }^{\rho -1}}{{({\iota }_1^{\rho }-{\theta }^{\rho })}^{1-\left({\zeta }_1\right)}}\right]d\theta \hfill \\ & +{\int }_{{\iota }_1}^{{\iota }_2}\frac{{\theta }^{\rho -1}}{{({\iota }_2^{\rho }-{\theta }^{\rho })}^{1-\left({\zeta }_1\right)}}d\theta |\hfill \\ & +|{\delta }_1\left({\iota }_2\right)-{\delta }_1\left({\iota }_1\right)|[{\widehat{\mathcal{N}}}_1\left\{\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left(\right|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right\}\hfill \\ & +\widehat{{\mathcal{N}}_2}\left\{|{\delta }_1|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\right\}\hfill \\ & +\left\{|{\varphi }_1|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\parallel x\parallel \hfill \\ & +\left\{|{\delta }_1||{\varphi }_2|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\parallel y\parallel ]\hfill \\ & \to 0as{\iota }_2\to {\iota }_1.\hfill \end{array}$$

(47)

independent of $(x,y)$ with respect to $|f(\iota ,x\left({\iota }_1\right),y\left({\iota }_1\right))|\le \widehat{{\mathcal{N}}_1}$ and $|g(\iota ,x\left({\iota }_1\right),y\left({\iota }_1\right))|\le \widehat{{\mathcal{N}}_2}$. Similarly, we can express $|{\mathsf{\Psi }}_2(x,y)\left({\iota }_2\right)-{\mathsf{\Psi }}_2(x,y)\left({\iota }_1\right)|\to 0$ as ${\iota }_2\to {\iota }_1$ independent of $(x,y)$ in terms of the boundedness of f and g. The operator $\mathsf{\Psi }$ is equicontinuous due to the equicontinuity of ${\mathsf{\Psi }}_1$ and ${\mathsf{\Psi }}_2$. The operator is compact as a result of the Arzela–Ascoli theorem. Finally, we show that the set $\mathsf{\Pi }\left(\mathsf{\Psi }\right)=\left\{\right(x,y)\in \mathcal{P}\times \mathcal{Q}:\lambda \mathsf{\Psi }(x,y);0<\lambda <1\}$ is bounded. Let $(x,y)\in \mathsf{\Pi }\left(\mathsf{\Psi }\right)$. Then, $(x,y)=\lambda \mathsf{\Psi }(x,y)$. For any $\iota \in \mathcal{E}$, we have $x\left(\iota \right)=\lambda {\mathsf{\Psi }}_1(x,y)\left(\iota \right),y\left(\iota \right)=\lambda {\mathsf{\Psi }}_2(x,y)\left(\iota \right)$. By utilizing (${\mathcal{A}}_1$) in (34), we obtain

$$\begin{array}{cc}\hfill \left|x\right(\iota \left)\right|& {\le }^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}({\psi }_0,{\psi }_1\left|x\left(\iota \right)\right|,{\psi }_2\left|y\left(\iota \right)\right|)+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\iota \right)\right|\hfill \\ & +|{\delta }_1(\iota )|[|{\mathcal{E}}_2\left|\right(\left|ϵ\right|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}(\widehat{{\psi }_0}+\widehat{{\psi }_1}|x\left({\varpi }_j\right)|+\widehat{{\psi }_2}|y\left({\varpi }_j\right)\left|\right)\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y\left({\varpi }_j\right)|+{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}({\psi }_0,{\psi }_1|x\left(\mathcal{S}\right)|,{\psi }_2|y\left(\mathcal{S}\right)\left|\right)+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\mathcal{S}\right)\right|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}({\psi }_0,{\psi }_1|x\left({\sigma }_j\right)|,{\psi }_2|y\left({\sigma }_j\right)\left|\right)+\left|\pi \right||{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left({\sigma }_j\right)\right|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}(\widehat{{\psi }_0}+\widehat{{\psi }_1}\left|x\left(\mathcal{S}\right)\right|+\widehat{{\psi }_2}\left|y\left(\mathcal{S}\right)\right|)+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}\left|y\left(\mathcal{S}\right)\right|\left)\right],\hfill \end{array}$$

which is obtained when the norm for $\iota \in \mathcal{E}$ is taken,

$$\begin{array}{c}\hfill \left|\right|x\left|\right|\le ({\psi }_0+{\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|){\mathcal{U}}_1+(\widehat{{\psi }_0}+\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|){\mathcal{V}}_1+\parallel x\parallel \widehat{{\mathcal{U}}_1}+\parallel y\parallel \widehat{{\mathcal{V}}_1}.\end{array}$$

(48)

Likewise, we have the ability to get

$$\begin{array}{c}\hfill \left|\right|y\left|\right|\le (\widehat{{\psi }_0}+\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|){\mathcal{V}}_2+({\psi }_0+{\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|){\mathcal{U}}_2+\parallel x\parallel \widehat{{\mathcal{U}}_2}+\parallel y\parallel \widehat{{\mathcal{V}}_2}.\end{array}$$

(49)

From (48) and (49), we get

$$\begin{array}{cc}\hfill \left|\right|x\left|\right|+\left|\right|y\left|\right|=& {\psi }_0({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_0}({\mathcal{V}}_1+{\mathcal{V}}_2)+\left|\right|x\left|\right|\left[{\psi }_1({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_1}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2}\right]\hfill \\ & +\left|\right|y\left|\right|\left[{\psi }_1({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_1}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}\right],\hfill \end{array}$$

which results, with $\left|\right|(x,y)\left|\right|=\left|\right|x\left|\right|+\left|\right|y\left|\right|$, in

$$\begin{array}{c}\hfill \left|\right|(x,y)\left|\right|\le \frac{{\psi }_0({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_0}({\mathcal{V}}_1+{\mathcal{V}}_2)}{\mathsf{\Phi }}.\end{array}$$

Thus, $\mathsf{\Pi }\left(\mathsf{\Psi }\right)$ is bounded. Hence the operator $\mathsf{\Psi }$ has a fixed point by Leray–Schauder’s nonlinear alternative [39], which corresponds to at least one solution of the problem in (3) and (4) on $\mathcal{E}$. □

 Theorem 2.

If the assumption $\left({\mathcal{A}}_2\right)$ is satisfied, then the problem in (3) and (4) has a unique solution on $\mathcal{E}$, and there exist ${\mathcal{S}}_1,{\mathcal{S}}_2>0$ such that $\left|f(\iota ,0,0)\right|\le {\mathcal{S}}_1,\left|g(\iota ,0,0)\right|\le {\mathcal{S}}_2,$ Then, given that

$$\begin{array}{c}\hfill ({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2})+(\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2})+(\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2})<1,\end{array}$$

(50)

where ${\mathcal{U}}_1,{\mathcal{V}}_1,\widehat{{\mathcal{U}}_1},\widehat{{\mathcal{V}}_1},{\mathcal{U}}_2,{\mathcal{V}}_2,\widehat{{\mathcal{U}}_2},\widehat{{\mathcal{V}}_2}$ are given by (36)–(43), respectively.

 Proof. 

Let us fix $\phi \le \frac{({\mathcal{U}}_1+{\mathcal{U}}_2){\mathcal{S}}_1+({\mathcal{V}}_1+{\mathcal{V}}_2){\mathcal{S}}_2}{1-(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2})+\widehat{{\mathcal{U}}_2})+(\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}))}$ and demonstrate that $\mathsf{\Psi }{\mathcal{B}}_{\phi }\subset {\mathcal{B}}_{\phi }$ when operator $\mathsf{\Psi }$ is given by (33) and ${\mathcal{B}}_{\phi }=\{(x,y)\in \mathcal{P}\times \mathcal{Q}:|\left|(x,y)\right||\le \phi \}$. For $(x,y)\in {\mathcal{B}}_{\phi },\iota \in \mathcal{E}$

$$\begin{array}{c}\hfill \left|f(\iota ,x\left(\iota \right),y\left(\iota \right))\right|\le {\psi }_1\left|x\left(\iota \right)\right|+{\psi }_2\left|y\left(\iota \right)\right|+{\mathcal{S}}_1\\ \hfill \le {\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|+{\mathcal{S}}_1,\end{array}$$

and

$$\begin{array}{c}\hfill \left|g(\iota ,x\left(\iota \right),y\left(\iota \right))\right|\le \widehat{{\psi }_1}\left|x\left(\iota \right)\right|+\widehat{{\psi }_2}\left|y\left(\iota \right)\right|+{\mathcal{S}}_2\\ \hfill \le \widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|+{\mathcal{S}}_2.\end{array}$$

(51)

This leads to

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_1(x,y)\left(\iota \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\iota ,x\left(\iota \right),y\left(\iota \right))-f(\iota ,0,0)|+\left|f(\iota ,0,0)\right|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\iota \right)\right|\hfill \\ & +|{\delta }_1(\iota )|[|{\mathcal{E}}_2\left|\right(\left|ϵ\right|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g\right(({\varpi }_j,x\left({\varpi }_j\right),y\left({\varpi }_j\right))-g({\varpi }_j,0,0)|+|g({\varpi }_j,0,0)\left|\right)\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y\left({\varpi }_j\right)|+{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))-f(\mathcal{S},0,0)|+|f(\mathcal{S},0,0)|\hfill \\ & +|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\mathcal{S}\right)\right|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f({\sigma }_j,x\left({\sigma }_j\right),y\left({\sigma }_j\right))-f({\sigma }_j,0,0)|+|f({\sigma }_j,0,0)|\hfill \\ & +\left|\pi \right||{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left({\sigma }_j\right)\right|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|g\right((\mathcal{S},x\left(\mathcal{S}\right),y\left(\mathcal{S}\right))-g(\mathcal{S},0,0)|+|g(\mathcal{S},0,0)\left|\right)+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}\left|y\left(\mathcal{S}\right)\right|\left)\right]\hfill \\ \hfill \le & ({\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|+{\mathcal{S}}_1)\left\{\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right\}\hfill \\ \hfill +& (\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|+{\mathcal{S}}_2)\left\{\left|{\delta }_1\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\right\}\hfill \\ \hfill +& \left\{\left|{\varphi }_1\right|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\parallel x\parallel \hfill \\ \hfill +& \left\{\left|{\delta }_1\left|\right|{\varphi }_2\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\parallel y\parallel \hfill \end{array}$$

$$\begin{array}{c}\hfill \left|\right|{\mathsf{\Psi }}_1(x,y)\left|\right|\le ({\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|+{\mathcal{S}}_1){\mathcal{U}}_1+(\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|+{\mathcal{S}}_2){\mathcal{V}}_1+\parallel x\parallel \widehat{{\mathcal{U}}_1}+\parallel y\parallel \widehat{{\mathcal{V}}_1}.\end{array}$$

(52)

Similarly, we obtain

$$\begin{array}{cc}\hfill |{\mathsf{\Psi }}_2(x,y)\left(\iota \right)|\le & (\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|+{\mathcal{S}}_2)\left\{\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right\}\hfill \\ & +({\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|+{\mathcal{S}}_1)\left\{\left|{\delta }_2\right|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right)\right\}\hfill \\ & +\left\{\left|{\varphi }_2\right|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\parallel y\parallel \hfill \\ & +\left\{\left|{\delta }_2\left|\right|{\varphi }_1\right|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\parallel x\parallel \hfill \\ \hfill \left|\right|{\mathsf{\Psi }}_2(x,y)\left|\right|\le & (\widehat{{\psi }_1}\left|\right|x\left|\right|+\widehat{{\psi }_2}\left|\right|y\left|\right|+{\mathcal{S}}_2){\mathcal{K}}_2+({\psi }_1\left|\right|x\left|\right|+{\psi }_2\left|\right|y\left|\right|+{\mathcal{S}}_1){\mathcal{J}}_2+\parallel x\parallel \widehat{{\mathcal{U}}_2}+\parallel y\parallel \widehat{{\mathcal{V}}_2}.\hfill \end{array}$$

(53)

As a result, (52) and (53) follow $\left|\right|\mathsf{\Psi }(x,y)\left|\right|\le \phi ,$ and thus $\mathsf{\Psi }{\mathcal{B}}_{\phi }\subset {\mathcal{B}}_{\phi }$. Now, for $(x_1,y_1),(x_2,y_2)\in \mathcal{P}\times \mathcal{Q}$ and any $\iota \in \mathcal{E}$, we get

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_1(x_1,y_1)\left(\iota \right)-{\mathsf{\Psi }}_1(x_2,y_2)\left(\iota \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\iota ,x_1\left(\iota \right),y_1\left(\iota \right))-f(\iota ,x_2\left(\iota \right),y_2\left(\iota \right))|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}|x_1\left(\iota \right)-x_2\left(\iota \right)|\hfill \\ & +|{\delta }_1\left(\iota \right)|[|{\mathcal{E}}_2|(|ϵ|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}|g({\varpi }_j,x_1\left({\varpi }_j\right),y_1\left({\varpi }_j\right))-g({\varpi }_j,x_2\left({\varpi }_j\right),y_2\left({\varpi }_j\right))|\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y_1\left({\varpi }_j\right)-y_2\left({\varpi }_j\right)|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f(\mathcal{S},x_1\left(\mathcal{S}\right),y_1\left(\mathcal{S}\right))-f(\mathcal{S},x_2\left(\mathcal{S}\right),y_2\left(\mathcal{S}\right))|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}|x_1\left(\mathcal{S}\right)-x_2\left(\mathcal{S}\right)|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|f({\sigma }_j,x_1\left({\sigma }_j\right),y_1\left({\sigma }_j\right))-f({\sigma }_j,x_2\left({\sigma }_j\right),y_2\left({\sigma }_j\right))|\hfill \\ & +\left|\pi \right||{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}|x_1\left({\sigma }_j\right)-x_2\left({\sigma }_j\right)|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}|g(\mathcal{S},x_1\left(\mathcal{S}\right),y_1\left(\mathcal{S}\right))-g(\mathcal{S},x_2\left(\mathcal{S}\right),y_2\left(\mathcal{S}\right))|+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y_1\left(\mathcal{S}\right)-y_2\left(\mathcal{S}\right)|\left)\right]\hfill \\ \hfill \le & ({\psi }_1\left|\right|x_1-x_2\left|\right|+{\psi }_2\left|\right|y_1-y_2||)\left\{\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right\}\hfill \\ & +(\widehat{{\psi }_1}\left|\right|x_1-x_2\left|\right|+\widehat{{\psi }_2}\left|\right|y_1-y_2\left|\right|)\left\{\left|{\delta }_1\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\right\}\hfill \\ & +\left\{\left|{\varphi }_1\right|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\left|\right|x_1-x_2\left|\right|\hfill \\ & +\left\{\left|{\delta }_1\left|\right|{\varphi }_2\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\left|\right|y_1-y_2\left|\right|\hfill \end{array}$$

$$\begin{array}{c}\hfill \le ({\mathcal{U}}_1({\psi }_1+{\psi }_2)+{\mathcal{V}}_1(\widehat{{\psi }_1}+\widehat{{\psi }_2})+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{V}}_1})\left(\right||x_1-x_2\left|\right|+\left|\right|y_1-y_2\left|\right|).\end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_2(x_1,y_1)\left(\iota \right)-{\mathsf{\Psi }}_2(x_2,y_2)\left(\iota \right)|\hfill \\ \hfill \le & (\widehat{{\psi }_1}\left|\right|x_1-x_2\left|\right|+\widehat{{\psi }_2}\left|\right|y_1-y_2\left|\right|)\left\{\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right\}\hfill \\ & +({\psi }_1\left|\right|x_1-x_2\left|\right|+{\psi }_2\left|\right|y_1-y_2\left|\right|)\left\{\left|{\delta }_2\right|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right)\right\}\hfill \\ & +\left\{\left|{\varphi }_2\right|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\right\}\left|\right|y_1-y_2\left|\right|\hfill \\ & +\left\{\left|{\delta }_2\left|\right|{\varphi }_1\right|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\right\}\left|\right|x_1-x_2\left|\right|\hfill \\ & \le ({\mathcal{U}}_2({\psi }_1+{\psi }_2)+{\mathcal{V}}_2(\widehat{{\psi }_1}+\widehat{{\psi }_2})+\widehat{{\mathcal{U}}_2}+\widehat{{\mathcal{V}}_2})\left(\right||x_1-x_2\left|\right|+\left|\right|y_1-y_2\left|\right|).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{c}\hfill \left|\right|{\mathsf{\Psi }}_1(x_1,y_1)\left(\iota \right)-{\mathsf{\Psi }}_1(x_2,y_2)\left(\iota \right)\left|\right|\le ({\mathcal{U}}_1({\psi }_1+{\psi }_2)+{\mathcal{V}}_1(\widehat{{\psi }_1}+\widehat{{\psi }_2})+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{V}}_1})\\ \hfill \left(\right||x_1-x_2||+||y_1-y_2|\left|\right).\end{array}$$

(54)

In a similar manner,

$$\begin{array}{c}\hfill \left|\right|{\mathsf{\Psi }}_2(x_1,y_1)\left(\iota \right)-{\mathsf{\Psi }}_2(x_2,y_2)\left(\iota \right)\left|\right|\le ({\mathcal{U}}_2({\psi }_1+{\psi }_2)+{\mathcal{V}}_2(\widehat{{\psi }_1}+\widehat{{\psi }_2})+\widehat{{\mathcal{U}}_2}+\widehat{{\mathcal{V}}_2})\\ \hfill \left(\right||x_1-x_2||+||y_1-y_2|\left|\right).\end{array}$$

(55)

Hence, using (54) and (55) we can get

$$\begin{array}{c}\hfill \left|\right|\mathsf{\Psi }(x_1,y_1)\left(\iota \right)-\mathsf{\Psi }(x_2,y_2)\left(\iota \right)\left|\right|\le (({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2})\\ \hfill +(\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2})+(\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2})\left)\right(\left|\right|x_1-x_2\left|\right|+\left|\right|y_1-y_2\left|\right|).\end{array}$$

As a consequence of condition $(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2})+(\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2})+(\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}))<1$, $\mathsf{\Psi }$ is a contraction operator. Hence, the operator has a unique fixed point by Banach’s contraction principle [39], which corresponds to a unique solution of the problem in (3) and (4). □

4. Example

Consider the following system of coupled generalized Liouville–Caputo type Langevin FDEs:

$$\left\{\begin{array}{c}{}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{63}{50}}({}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{3}{5}}+\frac{1}{100})x\left(\iota \right)=f(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,1],\hfill \\ {}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{44}{25}}({}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{17}{20}}+\frac{1}{150})y\left(\iota \right)=g(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,1],\hfill \end{array}\right.$$

(56)

augmented with boundary conditions

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=0,y\left(0\right)=0,x\left(1\right)=\frac{9}{50}{\sum }_{j=1}^k{\varsigma }_{j}y\left({\varpi }_j\right),y\left(1\right)=\frac{4}{25}{\sum }_{j=1}^k{\varrho }_{j}x\left({\sigma }_j\right),\hfill \end{array}\right.\end{array}$$

(57)

where ${\xi }_1=\frac{63}{50}$, ${\xi }_2=\frac{44}{25}$, ${\zeta }_1=\frac{3}{5}$, ${\zeta }_2=\frac{17}{20}$, $\rho =\frac{39}{50}$, $\mathcal{S}=1$, $ϵ=\frac{9}{50}$, $\pi =\frac{4}{25}$, ${\varpi }_1=\frac{8}{25}$, ${\varpi }_2=\frac{21}{50}$, ${\varpi }_3=\frac{13}{25}$, ${\varpi }_4=\frac{31}{50}$, ${\sigma }_1=\frac{7}{20}$, ${\sigma }_2=\frac{9}{20}$, ${\sigma }_3=\frac{11}{20}$, ${\sigma }_4=\frac{13}{20}$, ${\varsigma }_1=\frac{6}{25}$, ${\varsigma }_2=\frac{17}{50}$, ${\varsigma }_3=\frac{11}{25}$, ${\varsigma }_4=\frac{12}{25}$, ${\varrho }_1=\frac{7}{50}$, ${\varrho }_2=\frac{4}{25}$, ${\varrho }_3=\frac{1}{5}$, ${\varrho }_4=\frac{11}{50}$, and

$$\begin{array}{c}\hfill f(\iota ,x\left(\iota \right),y\left(\iota \right))=\frac{(\iota +1)}{200}\left(\frac{\left|y\right(\iota \left)\right|}{1+\left|y\right(\iota \left)\right|}+\frac{1}{4}cos\left(x\left(\iota \right)\right)+5\iota \right),\end{array}$$

(58)

$$\begin{array}{c}\hfill g(\iota ,x\left(\iota \right),y\left(\iota \right))=\frac{e^{-\iota }}{100}\left(\frac{1+\sqrt{\iota }}{4}+\frac{1}{7}cos\left(y\left(\iota \right)\right)+\frac{\left|x\right(\iota \left)\right|}{3(1+|x\left(\iota \right)\left|\right)}\right).\end{array}$$

(59)

with ${\psi }_0=\frac{1}{40},{\psi }_1=\frac{1}{800},{\psi }_2=\frac{1}{200},\widehat{{\psi }_0}=\frac{1}{400},\widehat{{\psi }_1}=\frac{1}{300}$, and $\widehat{{\psi }_2}=\frac{1}{700}$, using condition $\left({\mathcal{A}}_1\right)$. Next, we find that ${\mathcal{U}}_1=7.17720698699361$, ${\mathcal{V}}_1=15.76278111144004$, ${\mathcal{U}}_2=3.062775121234468$, ${\mathcal{V}}_2=17.146905979118152$, $\widehat{{\mathcal{U}}_1}=0.06427770462505382$, $\widehat{{\mathcal{V}}_1}=0.12551798367318132$, $\widehat{{\mathcal{U}}_2}=0.030350664250901854$, $\widehat{{\mathcal{V}}_2}=0.1356913209638917$, ${\mathcal{U}}_i,{\mathcal{V}}_i,\widehat{{\mathcal{U}}_i},\widehat{{\mathcal{V}}_i}(i=1,2)$ are, respectively, given by (36)–(43). Thus, ${\psi }_1({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_1}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2}\approxeq 0.21712730347976808<1,{\psi }_2({\mathcal{U}}_1+{\mathcal{U}}_2)+\widehat{{\psi }_2}({\mathcal{V}}_1+{\mathcal{V}}_2)+\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2}\approxeq 0.35942305387901086<1$, and there is at least one solution for the problem in (56) and (57) on $[0,1]$ with f and g, which are, respectively, given by (58) and (59). Additionally, all of the requirements of Theorem 1 have been satisfied.

Moreover, we employ

$$\begin{array}{c}\hfill f(\iota ,x\left(\iota \right),y\left(\iota \right))=\frac{\iota }{20}+\frac{1}{500}\frac{\left|y\right(\iota \left)\right|}{1+\left|y\right(\iota \left)\right|}+\frac{3}{800}cos\left(x\left(\iota \right)\right),\end{array}$$

(60)

$$\begin{array}{c}\hfill g(\iota ,x\left(\iota \right),y\left(\iota \right))=\frac{(e^{-\iota }+1)}{50}+\frac{1}{400}cos\left(y\left(\iota \right)\right)+\frac{1}{700}\frac{\left|x\right(\iota \left)\right|}{1+\left|x\right(\iota \left)\right|},\end{array}$$

(61)

to illustrate Theorem 2. Using assumption $\left({\mathcal{A}}_2\right)$ with ${\psi }_1=\frac{3}{800},{\psi }_2=\frac{1}{500},\widehat{{\psi }_1}=\frac{1}{700}$, and $\widehat{{\psi }_2}=\frac{1}{400}$. The assumptions of Theorem 2 are also satisfied with $({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2})+(\widehat{{\mathcal{U}}_1}+\widehat{{\mathcal{U}}_2})+(\widehat{{\mathcal{V}}_1}+\widehat{{\mathcal{V}}_2})\approxeq 0.5440056270625331<1$. Consequently, there exists a unique solution on $[0,1]$ to the problem in (56) and (57) with f and g supplied by (60) and (62), respectively, according to Theorem 2.

For brevity, we use these notations:

$$\begin{array}{c}\hfill {\mathcal{U}}_1=\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\\ \hfill +|{\varphi }_1|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right),\end{array}$$

(62)

$$\begin{array}{c}\hfill {\mathcal{V}}_1=\left|{\delta }_1\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\\ \hfill +|{\delta }_1\left|\right|{\varphi }_2|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right),\end{array}$$

(63)

$$\begin{array}{c}\hfill {\mathcal{U}}_2=\left|{\delta }_2\right|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right)\\ \hfill +|{\delta }_2\left|\right|{\varphi }_1|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right),\end{array}$$

(64)

$$\begin{array}{c}\hfill {\mathcal{V}}_2=\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\\ \hfill +|{\varphi }_2|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right),\end{array}$$

(65)

5. Ulam–Hyers Stability Results

With the help of an integral formulation of the solution provided by

$$\begin{array}{c}\hfill x\left(\iota \right)={\mathsf{\Psi }}_1(x,y)\left(\iota \right),y\left(\iota \right)={\mathsf{\Psi }}_2(x,y)\left(\iota \right),\end{array}$$

(66)

where ${\mathsf{\Psi }}_1$ and ${\mathsf{\Psi }}_2$ are given by (34) and (35), we analyse the Ulam–Hyers stability for problem (3) in this section. Consider the following definitions of nonlinear operators

$$\begin{array}{c}\hfill {\mathcal{H}}_1,{\mathcal{H}}_2\in \mathcal{C}(\mathcal{E},\mathbb{R})\times \mathcal{C}(\mathcal{E},\mathbb{R})\to \mathcal{C}(\mathcal{E},\mathbb{R}),\end{array}$$

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_1}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right)-f(\iota ,x\left(\iota \right),y\left(\iota \right))={\mathcal{H}}_1(x,y)\left(\iota \right),\iota \in \mathcal{E},\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_2}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right)-g(\iota ,x\left(\iota \right),y\left(\iota \right))={\mathcal{H}}_1(x,y)\left(\iota \right),\iota \in \mathcal{E}.\hfill \end{array}\right.$$

For some $\widehat{{\lambda }_1},{\widehat{\lambda }}_2>0$, taking into consideration the following inequality

$$\begin{array}{c}\hfill \left|\right|{\mathcal{H}}_1(x,y)\left|\right|\le \widehat{{\lambda }_1},\left|\right|{\mathcal{H}}_2(x,y)\left|\right|\le \widehat{{\lambda }_2},\end{array}$$

(67)

 Definition 4.

The system in (3) and (4) is UHS if ${\mathcal{V}}_1,{\mathcal{V}}_2>0$ and ∃ a unique solution $(x,y)\in \mathcal{C}(\mathcal{E},\mathbb{R})$ of the problem in (3) and (4) with

$$\begin{array}{c}\hfill \left|\right|(x,y)-(x^{*},y^{*})\left|\right|\le {\mathcal{V}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_2\widehat{{\lambda }_2},\end{array}$$

$\forall (x,y)\in \mathcal{C}(\mathcal{E},\mathbb{R})$ of inequality (67).

 Theorem 3.

If assumption $\left({\mathcal{A}}_2\right)$ is satisfied, then the BVP (3) and (4) is UHS.

 Proof. 

Let $(x,y)\in \mathcal{C}(\mathcal{E},\mathbb{R})\times \mathcal{C}(\mathcal{E},\mathbb{R})$ be the solution of the BVP (3) and (4) satisfying (34) and (35). Let $(x,y)$ be any solution satisfying (67):

$$\left\{\begin{array}{c}{}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_1}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right)=f(\iota ,x\left(\iota \right),y\left(\iota \right))+{\mathcal{H}}_1(x,y)\left(\iota \right),\iota \in \mathcal{E},\hfill \\ {}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\xi }_2}({}_C^{\rho }{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right)=g(\iota ,x\left(\iota \right),y\left(\iota \right))+{\mathcal{H}}_1(x,y)\left(\iota \right),\iota \in \mathcal{E},\hfill \end{array}\right.$$

thus,

$$\begin{array}{cc}\hfill x^{*}\left(\iota \right)& ={\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right){+}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}{\mathcal{H}}_1(x,y)\left(\iota \right)-{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\iota \right)\hfill \\ & +{\delta }_1\left(\iota \right)\left[{\mathcal{E}}_2\right(ϵ\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}{\mathcal{H}}_2(x,y)\left({\varpi }_j\right)\hfill \\ & -ϵ{\varphi }_2\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left({\varpi }_j\right)-{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}{\mathcal{H}}_1(x,y)\left(\mathcal{S}\right)+{\varphi }_1^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left(\mathcal{S}\right))\hfill \\ & +{\mathcal{E}}_1(\pi \sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}{\mathcal{H}}_1(x,y)\left({\sigma }_j\right)-\pi {\varphi }_1\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}x\left({\sigma }_j\right)\hfill \\ & -{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}{\mathcal{H}}_2(x,y)\left(\mathcal{S}\right)+{\varphi }_2^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}y\left(\mathcal{S}\right)\left)\right].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right)-x^{*}\left(\iota \right)|\hfill \\ \hfill \le & {}^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|{\mathcal{H}}_1(x,y)\left(\iota \right)|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\iota \right)\right|\hfill \\ & +|{\delta }_1(\iota )|[|{\mathcal{E}}_2|(|ϵ|\sum _{j=1}^k{\varsigma }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}\left|{\mathcal{H}}_2(x,y)\left({\varpi }_j\right)\right|\hfill \\ & +\left|ϵ\right||{\varphi }_2|\sum _{j=1}^k{\varsigma }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_2}|y\left({\varpi }_j\right)|+{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}|{\mathcal{H}}_1(x,y)\left(\mathcal{S}\right)|+|{\varphi }_1{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left(\mathcal{S}\right)\right|)\hfill \\ & +|{\mathcal{E}}_1\left|\right(\left|\pi \right|\sum _{j=1}^k{\varrho }_j^{\rho }{\mathcal{I}}_{0^{+}}^{{\xi }_1+{\zeta }_1}{\mathcal{H}}_1(x,y)\left({\sigma }_j\right)+\left|\pi \right||{\varphi }_1|\sum _{j=1}^k{\varrho }_j{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\zeta }_1}\left|x\left({\sigma }_j\right)\right|\hfill \\ & +{{}^{\rho }\mathcal{I}}_{0^{+}}^{{\xi }_2+{\zeta }_2}|{\mathcal{H}}_2(x,y)\left(\mathcal{S}\right)|+|{\varphi }_2{|}^{\rho }{\mathcal{I}}_{0^{+}}^{{\zeta }_2}\left|y\left(\mathcal{S}\right)\right|\left)\right]\hfill \\ & \le {\widehat{\lambda }}_1\{\frac{\left({\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\hfill \\ & +|{\varphi }_1|\left(\frac{\left({\mathcal{S}}^{\rho {\zeta }_1}(1+|{\delta }_1\left|\right|{\mathcal{E}}_2\left|\right)\right)}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{|{\delta }_1\left|\right|\pi \left|\right|{\mathcal{E}}_1|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\}\hfill \\ & +{\widehat{\lambda }}_2\{\left|{\delta }_1\right|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\right)\hfill \\ & +|{\delta }_1\left|\right|{\varphi }_2|\left(\frac{|{\mathcal{E}}_1|{\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\mathcal{E}}_2\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\}\hfill \\ \hfill \le & {\mathcal{U}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_1\widehat{{\lambda }_2}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}\hfill & |{\mathsf{\Psi }}_2(x^{*},y^{*})\left(\iota \right)-y^{*}\left(\iota \right)|\hfill \\ \hfill \le & \widehat{{\lambda }_2}\{\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho ({\xi }_2+{\zeta }_2)}}{{\rho }^{{\xi }_2+{\zeta }_2}\mathsf{\Gamma }({\xi }_2+{\zeta }_2+1)}\hfill \\ & +|{\varphi }_2|\left(\frac{(1+|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_1}\left|\right){\mathcal{S}}^{\rho {\zeta }_2}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}+\frac{|{\delta }_2\left|\right|\widehat{{\mathcal{E}}_2}\left|\right|ϵ|{\sum }_{j=1}^k{\varsigma }_j{\varpi }_j^{\rho \left({\zeta }_2\right)}}{{\rho }^{{\zeta }_2}\mathsf{\Gamma }({\zeta }_2+1)}\right)\}\hfill \\ & +\widehat{{\lambda }_1}\{\left|{\delta }_2\right|\left(\frac{{\mathcal{S}}^{\rho ({\xi }_1+{\zeta }_1)}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho ({\xi }_1+{\zeta }_1)}}{{\rho }^{{\xi }_1+{\zeta }_1}\mathsf{\Gamma }({\xi }_1+{\zeta }_1+1)}\right)\hfill \\ & +|{\delta }_2\left|\right|{\varphi }_1|\left(\frac{{\mathcal{S}}^{\rho {\zeta }_1}\left|\widehat{{\mathcal{E}}_2}\right|}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}+\frac{\left|\pi \right||\widehat{{\mathcal{E}}_1}|{\sum }_{j=1}^k{\varrho }_j{\sigma }_j^{\rho \left({\zeta }_1\right)}}{{\rho }^{{\zeta }_1}\mathsf{\Gamma }({\zeta }_1+1)}\right)\}\hfill \\ \hfill \le & {\mathcal{U}}_2{\widehat{\lambda }}_1+{\mathcal{V}}_2{\widehat{\lambda }}_2,\hfill \end{array}$$

where ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{U}}_2$, and ${\mathcal{V}}_2$ are defined in (62)–(65), respectively. Consequently, based on the fixed-point property of the operator $\mathsf{\Psi }$, provided in (34) and (35), we derive

$$\begin{array}{cc}\hfill |x\left(\iota \right)-x^{*}\left(\iota \right)|=& |x\left(\iota \right)-{\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right)+{\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right)-x^{*}\left(\iota \right)|\hfill \\ \hfill \le & |{\mathsf{\Psi }}_1(x,y)\left(\iota \right)-{\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right)|+|{\mathsf{\Psi }}_1(x^{*},y^{*})\left(\iota \right)-x^{*}\left(\iota \right)|\hfill \\ \hfill \le & (({\mathcal{U}}_1{\psi }_1+{\mathcal{V}}_1\widehat{{\psi }_1})+({\mathcal{U}}_1{\psi }_2+{\mathcal{V}}_1\widehat{{\psi }_2}))\left|\right|(x,y)-(x^{*},y^{*})\left|\right|\hfill \\ & +{\mathcal{U}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_1\widehat{{\lambda }_2}.\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill |y\left(\iota \right)-y^{*}\left(\iota \right)|=& |y\left(\iota \right)-{\mathsf{\Psi }}_2(x^{*},y^{*})\left(\iota \right)+{\mathsf{\Psi }}_2(x^{*},y^{*})\left(\iota \right)-y^{*}\left(\iota \right)|\hfill \\ \hfill \le & |{\mathsf{\Psi }}_2(x,y)\left(\iota \right)-{\mathsf{\Psi }}_2(x^{*},y^{*})\left(\iota \right)|+|{\mathsf{\Psi }}_2(x^{*},y^{*})\left(\iota \right)-y^{*}\left(\iota \right)|\hfill \\ \hfill \le & (({\mathcal{U}}_2{\psi }_1+{\mathcal{V}}_2\widehat{{\psi }_1})+({\mathcal{U}}_2{\psi }_2+{\mathcal{V}}_2\widehat{{\psi }_2}))\left|\right|(x,y)-(x^{*},y^{*})\left|\right|\hfill \\ & +{\mathcal{U}}_2\widehat{{\lambda }_1}+{\mathcal{V}}_2\widehat{{\lambda }_2}.\hfill \end{array}$$

(69)

From the above equations (68) and (69) it follows that

$$\begin{array}{cc}\hfill \left|\right|(x,y)-(x^{*},y^{*})\left|\right|\le & ({\mathcal{U}}_1+{\mathcal{U}}_2)\widehat{{\lambda }_1}+({\mathcal{V}}_1+{\mathcal{V}}_2)\widehat{{\lambda }_2}\hfill \\ & +(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2}))\left|\right|(x,y)-(x^{*},y^{*})\left|\right|.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left|\right|(x,y)-(x^{*},y^{*})\left|\right|\le & \frac{({\mathcal{U}}_1+{\mathcal{U}}_2)\widehat{{\lambda }_1}+({\mathcal{V}}_1+{\mathcal{V}}_2)\widehat{{\lambda }_2}}{1-(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2}))}\hfill \\ \hfill \le & {\mathcal{V}}_1\widehat{{\lambda }_1}+{\mathcal{V}}_2\widehat{{\lambda }_2},\hfill \end{array}$$

with

$$\begin{array}{c}\hfill {\mathcal{V}}_1=\frac{{\mathcal{U}}_1+{\mathcal{U}}_2}{1-(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2}))},\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{V}}_2=\frac{{\mathcal{V}}_1+{\mathcal{V}}_2}{1-(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2}))}.\end{array}$$

Thus, the BVP (3) and (4) is UHS. □

6. Example

Consider the following system of coupled generalized Liouville–Caputo type Langevin FDEs:

$$\left\{\begin{array}{c}{}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{63}{50}}({}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{3}{5}}+\frac{1}{100})x\left(\iota \right)=\frac{\sqrt{\iota }}{30}+\frac{1}{40(16+\iota )}\frac{\left|y\right(\iota \left)\right|}{1+\left|y\right(\iota \left)\right|}+\frac{4}{900}cos\left(x\left(\iota \right)\right),\iota \in [0,1],\hfill \\ {}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{44}{25}}({}_C^{\frac{39}{50}}{\mathcal{D}}_{0+}^{\frac{17}{20}}+\frac{1}{150})y\left(\iota \right)=\frac{\iota }{70}+\frac{1}{400}cos\left(y\left(\iota \right)\right)+\frac{1}{800}\frac{\left|x\right(\iota \left)\right|}{1+\left|x\right(\iota \left)\right|},\iota \in [0,1],\hfill \end{array}\right.$$

(70)

augmented with boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}x\left(0\right)=0,y\left(0\right)=0,x\left(1\right)=\frac{9}{50}{\sum }_{j=1}^k{\varsigma }_{j}y\left({\varpi }_j\right),y\left(1\right)=\frac{4}{25}{\sum }_{j=1}^k{\varrho }_{j}x\left({\sigma }_j\right),\hfill \end{array}\right.\end{array}$$

(71)

where ${\xi }_1=\frac{63}{50}$, ${\xi }_2=\frac{44}{25}$, ${\zeta }_1=\frac{3}{5}$, ${\zeta }_2=\frac{17}{20}$, $\rho =\frac{39}{50}$, $\mathcal{S}=1$, $ϵ=\frac{9}{50}$, $\pi =\frac{4}{25}$, ${\varpi }_1=\frac{8}{25}$, ${\varpi }_2=\frac{21}{50}$, ${\varpi }_3=\frac{13}{25}$, ${\varpi }_4=\frac{31}{50}$, ${\sigma }_1=\frac{7}{20}$, ${\sigma }_2=\frac{9}{20}$, ${\sigma }_3=\frac{11}{20}$, ${\sigma }_4=\frac{13}{20}$, ${\varsigma }_1=\frac{6}{25}$, ${\varsigma }_2=\frac{17}{50}$, ${\varsigma }_3=\frac{11}{25}$, ${\varsigma }_4=\frac{12}{25}$, ${\varrho }_1=\frac{7}{50}$, ${\varrho }_2=\frac{4}{25}$, ${\varrho }_3=\frac{1}{5}$, ${\varrho }_4=\frac{11}{50}$ and

$$\begin{array}{c}\hfill |f(\iota ,x_1\left(\iota \right),y_1\left(\iota \right))-f(\iota ,x_2\left(\iota \right),y_2\left(\iota \right))|=\frac{4}{900}|x_1\left(\iota \right)-x_2\left(\iota \right)|+\frac{1}{640}|y_1\left(\iota \right)-y_2\left(\iota \right)|,\end{array}$$

(72)

$$\begin{array}{c}\hfill |g(\iota ,x_1\left(\iota \right),y_1\left(\iota \right))-g(\iota ,x_2\left(\iota \right),y_2\left(\iota \right))|=\frac{1}{800}|x_1\left(\iota \right)-x_2\left(\iota \right)|+\frac{1}{400}|y_1\left(\iota \right)-y_2\left(\iota \right)|.\end{array}$$

(73)

with ${\psi }_1=\frac{4}{900},{\psi }_2=\frac{1}{640},\widehat{{\psi }_1}=\frac{1}{800}$, and $\widehat{{\psi }_2}=\frac{1}{400}$, and using condition $\left({\mathcal{A}}_2\right)$. Next, we find that ${\mathcal{U}}_1=7.241484691618664$, ${\mathcal{V}}_1=15.88829909511322$, ${\mathcal{U}}_2=3.0931257854853698$, ${\mathcal{V}}_2=17.282597300082045$, ${\mathcal{U}}_i$, ${\mathcal{V}}_i$ are, respectively, given by (62)–(65). Thus, $(({\mathcal{U}}_1+{\mathcal{U}}_2)({\psi }_1+{\psi }_2)+({\mathcal{V}}_1+{\mathcal{V}}_2)(\widehat{{\psi }_1}+\widehat{{\psi }_2}))\approxeq 0.18647029247291969<1$. Consequently, there exists a unique solution on $[0,1]$, to the problem in (70) and (71), which is stable for Ulam–Hyers, with f and g supplied by (72) and (73), respectively, according to Theorem 3.

7. Asymmetric Case

 Remark 1.

If $\rho =1$, the problem (3)’s generalized Langevin FDEs reduces to the Caputo Langevin FDEs.

$$\left\{\begin{array}{c}{}_C{\mathcal{D}}_{0^{+}}^{{\xi }_1}({}_C{\mathcal{D}}_{0^{+}}^{{\zeta }_1}+{\varphi }_1)x\left(\iota \right)=f(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,\mathcal{S}],\hfill \\ {}_C{\mathcal{D}}_{0^{+}}^{{\xi }_2}({}_C{\mathcal{D}}_{0^{+}}^{{\zeta }_2}+{\varphi }_2)y\left(\iota \right)=g(\iota ,x\left(\iota \right),y\left(\iota \right)),\iota \in \mathcal{E}:=[0,\mathcal{S}].\hfill \end{array}\right.$$

(74)

8. Conclusions

We discussed the existence and uniqueness of solutions for a Langevin coupled system of fractional order involving generalised Liouville–Caputo type and multipoint boundary conditions in our contribution. To get at our result, we used the Leray–Schauder and Banach fixed-point theorems, and we included examples to help explain our study results. By using a conventional functional analysis, we demonstrated Ulam–Hyers stability. Our findings in this context are original and contribute to the body of knowledge on generalised fractional integral operators that are used to resolve generalised fractional differential equations of coupled Langevin systems with nonlocal multipoint boundary conditions. We highlighted the topic’s asymmetries in the remarks. The form of the solution in these kinds of statements can be used to conduct additional research on the positive solution and its asymmetry.