On the Existence of (p,q)-Solutions for the Post-Quantum Langevin Equation: A Fixed-Point-Based Approach

Abstract

The two-parameter $(p,q)$-operators are a new family of operators in calculus that have shown their capabilities in modeling various systems in recent years. Following this path, in this paper, we present a new construction of the Langevin equation using two-parameter $(p,q)$-Caputo derivatives. For this new Langevin equation, equivalently, we obtain the solution structure as a post-quantum integral equation and then conduct an existence analysis via a fixed-point-based approach. The use of theorems such as the Krasnoselskii and Leray–Schauder fixed-point theorems will guarantee the existence of solutions to this equation, whose uniqueness is later proven by Banach’s contraction principle. Finally, we provide three examples in different structures and validate the results numerically.

1. Introduction

By upgrading the classical differentiation and integration operators from integer orders to arbitrary non-integer orders, we arrive at fractional-order differential and integral calculus (shortly, fractional calculus), in which the order of all the operators can be fractional or complex numbers [1,2,3]. This field of applied mathematics, which originated in the early 17th century, has widely demonstrated its power and efficiency in various fields of science with extraordinary speed in recent decades. Classical operators can only analyze the local behavior of functions, while fractional-order operators, especially fractional-order derivatives, are inherently non-local in nature and can easily analyze systems that have memory and historical effects. This feature leads to the fact that these operators show their importance in modeling a variety of complex phenomena.

More precisely, considering these applications, we can refer to some published articles in this field. For example, we can mention the use of fractional-order differential equations in modeling the memory behavior of biological tissues and polymers, which was published by Lizama et al. in 2022 [4]. Fractional-order viscoelasticity equations can be used to describe non-integer damping [5]. Fractional-order diffusion equations can be used to analyze the type of particle motion in heterogeneous environments, such as diffusion in cellular tissues [6]. We can even see traces of fractional-order equations in the fields of medicine and epidemiology [7,8]. Impulsive fractional equations are other types of fractional equations in which we can analyze the dynamics of the solutions [9]. It is notable that the main tool for proving the existence of solutions for all the above models is based on fixed-point theory. In fact, most boundary or initial value problems involving fractional differential equations can be reformulated as fixed-point problems for operators acting on function spaces. Fixed-point theorems such as the Banach fixed-point theorem, Krasnoselskii fixed-point theorem, Schauder fixed-point theorem, etc., allow mathematicians to guarantee that solutions to fractional differential equations exist without necessarily constructing them explicitly. This is our approach in this paper.

q-calculus, which is the short term of quantum calculus, has its history dating back to the early 19th century and has achieved great comprehensiveness during these decades [10]. Of course, much earlier, Euler conducted initial studies on quantum series [11], and then Jacobi introduced elliptic functions and the theta function [12], which were very close to the same concept of quantum series. Later, in the early 19th century, Abel developed the binomial series and its expansions in the context of quantum calculus [13], and, finally, Jackson [14,15] systematically formalized quantum calculus. He defined the derivative and integral operators without the concept of limit in such a way that the parameter q played a fundamental role in their structure.

$(p,q)$-calculus, or, completely, post-quantum calculus, was extended after q-calculus by using two parameters, $0<q<p\le 1$. The $(p,q)$-operators are a modification or generalization of standard fractional operators by introducing additional parameters p and q that provide more flexibility in modeling complex systems. The first work is present in a paper published in 1991 by Chakrabarti et al. [16]. This calculus was developed and arranged by Sadjang [17] in 2018 systematically. Sadjang defined very important concepts of $(p,q)$-calculus, such as $(p,q)$-derivative, $(p,q)$-integral, $(p,q)$-Taylor expansion, and $(p,q)$-Laplace transform [18]. Recently, some researchers have studied the existence theory and stability analysis for different $(p,q)$-difference equations under a variety of fractional $(p,q)$-operators [19,20,21,22,23].

In 2020, Soontharanon et al. [24] formulated the Robin $(p,q)$-conditions for an ${\mathsf{\alpha }}_0$-order $(p,q)$-integro-difference equation

$$\left\{\begin{array}{c}{}^{RL}D_{p,q}^{{\mathsf{\alpha }}_0}\mathsf{\omega }\left(t\right)=g\left(t,\mathsf{\omega }\left(t\right),{}^{RL}I_{p,q}^{{\mathsf{\alpha }}_1}\left(f\mathsf{\omega }\right)\left(t\right),{}^{RL}D_{p,q}^{{\mathsf{\alpha }}_2}\mathsf{\omega }\left(t\right)\right),{\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2\in (0,1],{\mathsf{\alpha }}_0\in (1,2],t\in {\mathbb{I}}_{p,q}^T,\hfill \\ {\mathsf{\delta }}_1\mathsf{\omega }\left(a\right)+{\mathsf{\delta }}_2{}^{RL}D_{p,q}^{{\mathsf{\alpha }}_3}\mathsf{\omega }\left(a\right)=g_1\left(\mathsf{\omega }\left(t\right)\right),{\mathsf{\alpha }}_3\in (0,1],{\mathsf{\delta }}_1,{\mathsf{\delta }}_2\in {\mathbb{R}}^{+},\hfill \\ {\mathsf{\zeta }}_1\mathsf{\omega }\left({\displaystyle \frac{T}{p}}\right)+{\mathsf{\zeta }}_2{}^{RL}D_{p,q}^{{\mathsf{\alpha }}_3}\mathsf{\omega }\left({\displaystyle \frac{T}{p}}\right)=g_2\left(\mathsf{\omega }\left(t\right)\right),{\mathsf{\zeta }}_1,{\mathsf{\zeta }}_2\in {\mathbb{R}}^{+},0<q<p\le 1,\hfill \end{array}\right.$$

and conducted the existence analysis for the mentioned $(p,q)$-problem so that $a\in {\mathbb{I}}_{p,q}^T-\left\{0,{\displaystyle \frac{T}{p}}\right\}$ and ${\mathbb{I}}_{p,q}^T:=\left\{{\displaystyle \frac{q^{i}T}{p^{i+1}}}:i\in {\mathbb{N}}_0\right\}\cup \left\{0\right\}$. Set $\mathcal{X}={\mathbb{R}}^3$. Then, $g:{\mathbb{I}}_{p,q}^T\times \mathcal{X}\to \mathbb{R}$ is nonlinear and continuous and $g_1,g_2:C({\mathbb{I}}_{p,q}^T,\mathbb{R})\to \mathbb{R}$ and $f\in C({\mathbb{I}}_{p,q}^T\times {\mathbb{I}}_{p,q}^T,[0,\infty ))$. Moreover, ${}^{RL}D_{p,q}^{{\mathsf{\alpha }}^{*}}$ denotes the ${\mathsf{\alpha }}^{*}$-order $(p,q)$-derivative such that ${\mathsf{\alpha }}^{*}={\mathsf{\alpha }}_j(j=0,2,3)$ (in the Riemann–Liouville setting).

Later, in 2022, Neang et al. [25] considered a nonlinear function like $g\in C\left(\right[0,T]\times \mathbb{R},\mathbb{R})$ with the simplified domain and studied the existence theorems in relation to the $(p,q)$-solutions of the Caputo $(p,q)$-difference problem

$$\left\{\begin{array}{c}{}^{C}D_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\left(t\right)=g\left(p^{\mathsf{\alpha }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }}t\right)\right),\mathsf{\alpha }\in (1,2],0\le t\le T,\hfill \\ {\mathsf{\delta }}_1\mathsf{\omega }\left(0\right)+{\mathsf{\delta }}_2{D}_{p,q}\mathsf{\omega }\left(0\right)={\mathsf{\delta }}_3,{\mathsf{\delta }}_1,{\mathsf{\delta }}_2,{\mathsf{\delta }}_3\in \mathbb{R},\hfill \\ {\mathsf{\delta }}_4\mathsf{\omega }\left(T\right)+{\mathsf{\delta }}_5{D}_{p,q}\mathsf{\omega }\left(pT\right)={\mathsf{\delta }}_6,{\mathsf{\delta }}_4,{\mathsf{\delta }}_5,{\mathsf{\delta }}_6\in \mathbb{R},0<q<p\le 1,\hfill \end{array}\right.$$

with the aforementioned Caputo-type $(p,q)$-derivatives.

Recently, Etemad et al. [26] discussed $(p,q)$-solutions of the sequential $(p,q)$-Navier difference inclusion

$$\left\{\begin{array}{c}{}^{C}D_{p,q}^{{\mathsf{\alpha }}_1}\left({}^{C}D_{p,q}^{{\mathsf{\alpha }}_2}\mathsf{\omega }\right)\left(t\right)\in G\left(t,\mathsf{\omega }\left(t\right),{}^{C}D_{p,q}^{{\mathsf{\alpha }}_2}\mathsf{\omega }\left(t\right)\right),t\in [0,{\displaystyle \frac{T}{p}}],p,q\in (0,1),\hfill \\ {\mathsf{\delta }}_1\mathsf{\omega }\left(0\right)={\mathsf{\delta }}_2\mathsf{\omega }\left(1\right)={\mathsf{\delta }}_3{}^{C}D_{p,q}^{{\mathsf{\alpha }}_2}\mathsf{\omega }\left(0\right)={\mathsf{\delta }}_4{}^{C}D_{p,q}^{{\mathsf{\alpha }}_2}\mathsf{\omega }\left(1\right)=0,\hfill \end{array}\right.$$

where ${\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2\in (1,2]$, and ${\mathsf{\delta }}_1,{\mathsf{\delta }}_2,{\mathsf{\delta }}_3,{\mathsf{\delta }}_4\in {\mathbb{R}}^{+}$. ${}^{C}D_{p,q}^{(·)}$ denotes the Caputo fractional $(p,q)$-derivative, and G is a multifunction.

In this paper, to clarify the main purposes of this study, we aim to provide a new two-parametric model of the Langevin equation in which the newly defined two-parametric fractional $(p,q)$-derivatives of the $(p,q)$-Caputo type are substituted into the classical integer-order Langevin equation. More precisely, we formulate this structure for the Langevin equation in the context of a fractional sequential two-term boundary value problem of a $(p,q)$-difference equation, given by

$$\left\{\begin{array}{c}{}^{C}D_{p,q}^{\mathsf{\alpha }}\left({}^{C}D_{p,q}^{\mathsf{\beta }}+\mathsf{\mu }\right)\mathsf{\omega }\left(t\right)=g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))+g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)),\hfill \\ \mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left(pT\right),{}^{C}D_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(0\right)={}^{C}D_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(pT\right),\hfill \end{array}\right.$$

(1)

where all the parameters are defined as follows:

$$\begin{array}{cc}\hfill & q\in (0,1),p\in (0,1],q<p,t\in [0,T],T\ge 1\hfill \\ \hfill & \mathsf{\alpha }\in (0,1],\mathsf{\beta }\in (0,1],\mathsf{\mu }\in \mathbb{R}\setminus \left\{0\right\},\hfill \end{array}$$

(2)

and $g_1,g_2\in C([0,T]\times \mathbb{R},\mathbb{R})$. The symbols ${}^{C}D_{p,q}^{\mathsf{\alpha }}$ and ${}^{C}D_{p,q}^{\mathsf{\beta }}$ are used to denote the fractional $(p,q)$-Caputo-type derivatives of two distinct orders, $\mathsf{\alpha }$ and $\mathsf{\beta }$, respectively. The purpose of this study is limited to the existence theory of the $(p,q)$-solutions of a new category of the $(p,q)$-type Langevin equation via several fixed-point-based approaches.

The following sections of the paper are as follows: Section 2 aims to recall some basic notions on $(p,q)$-calculus. Section 3 is about the main existence theorems by using the well-known fixed-point methods. Section 4 illustrates the validated examples on the basis of the main existence theorems. The paper closes with Section 5 by stating the conclusions.

2. Preliminaries

In this section, we are aiming to provide several definitions and properties on $(p,q)$-calculus as a reminder. Throughout the paper, $q\in (0,1)$ and $0<q<p\le 1$.

The $(p,q)$-numbers play an important role in $(p,q)$-calculus and generalize q-numbers and ordinary numbers by depending them on two parameters, p and q. For each real number $\mathsf{\alpha }$, a $(p,q)$-number of $\mathsf{\alpha }$ is defined by

$${\displaystyle {\left[\mathsf{\alpha }\right]}_{p,q}=p^{\mathsf{\alpha }-1}{\left[\mathsf{\alpha }\right]}_{\frac{q}{p}}=\frac{p^{\mathsf{\alpha }}-q^{\mathsf{\alpha }}}{p-q}.}$$

It is natural that ${\left[\mathsf{\alpha }\right]}_{p,q}\to \mathsf{\alpha }$ as $p=1$ and $q\to 1$ [27]. Moreover, ${\left[0\right]}_{p,q}=0$.

In the same way, the $(p,q)$-power functions generalize the ordinary power functions by depending them on two parameters, p and q. For each ${\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2\in \mathbb{R}$, we define

$${({\mathsf{\alpha }}_1-{\mathsf{\alpha }}_2)}_{p,q}^{\left(r\right)}={\mathsf{\alpha }}_1^r\prod _{j=0}^{\infty }{\displaystyle \frac{1}{p^r}}\left({\displaystyle \frac{1-\left(\frac{{\mathsf{\alpha }}_2}{{\mathsf{\alpha }}_1}\right){\left(\frac{q}{p}\right)}^j}{1-\left(\frac{{\mathsf{\alpha }}_2}{{\mathsf{\alpha }}_1}\right){\left(\frac{q}{p}\right)}^{r+j}}}\right),r\in \mathbb{R},{\mathsf{\alpha }}_1\ne 0,$$

(3)

and ${({\mathsf{\alpha }}_1-{\mathsf{\alpha }}_2)}_{p,q}^{\left(0\right)}=1$ [27]. Moreover, ${\left({\mathsf{\alpha }}_1\right)}_{p,q}^{\left(r\right)}={\displaystyle \frac{1}{p^r}}{\mathsf{\alpha }}_1^r$ when ${\mathsf{\alpha }}_2=0$ in (3).

The $(p,q)$-power function helps us to define the $(p,q)$-Gamma function as follows

$${\Gamma }_{p,q}\left(\mathsf{\alpha }\right)={\displaystyle \frac{{(p-q)}_{p,q}^{(\mathsf{\alpha }-1)}}{{(p-q)}^{\mathsf{\alpha }-1}}},\mathsf{\alpha }\ne {\mathbb{Z}}^{-}.$$

Evidently, ${\Gamma }_{p,q}(\mathsf{\alpha }+1)={\left[\mathsf{\alpha }\right]}_{p,q}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)$ [27].

A $(p,q)$-derivative is a special form of derivative that depends on the parameters p and q, and it uses a difference structure instead of a limit structure. In other words, a $(p,q)$-derivative of the function $\mathsf{\omega }$ is defined as $\left(D_{p,q}\mathsf{\omega }\right)\left(0\right)={lim}_{t\to 0}\left(D_{p,q}\mathsf{\omega }\right)\left(t\right)$ and

$$\left(D_{p,q}\mathsf{\omega }\right)\left(t\right)={\displaystyle \frac{\mathsf{\omega }\left(pt\right)-\mathsf{\omega }\left(qt\right)}{(p-q)t}},$$

for $t\ne 0$ [27].

For example, if $\mathsf{\omega }\left(t\right)=t^2$, then, since ${\left[2\right]}_{p,q}=p+q$, we have

$$D_{p,q}{t}^2={\displaystyle \frac{p^2{t}^2-q^2{t}^2}{(p-q)t}}={\displaystyle \frac{(p^2-q^2)t^2}{(p-q)t}}=(p+q)t={\left[2\right]}_{p,q}t.$$

As we know, extending ordinary operators to fractional operators is a useful way that broadens the scope of calculus to describe all complex phenomena adequately. Hence, we recall fractional versions of two $(p,q)$-operators [27].

Assume that $\mathsf{\alpha }>0$ and $\mathsf{\omega }\left(t\right)$ is a function. The fractional Riemann–Liouville (RL) $(p,q)$-integral ${}^{RL}I_{p,q}^{\mathsf{\alpha }}$ for the function $\mathsf{\omega }$ is defined as

$$({}^{RL}I_{p,q}^{\mathsf{\alpha }}\mathsf{\omega })\left(t\right)={\int }_0^t{\displaystyle \frac{{(t-qs)}_{p,q}^{(\mathsf{\alpha }-1)}}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}\mathsf{\omega }\left(p^{1-\mathsf{\alpha }}s\right){\mathrm{d}}_{p,q}s,$$

with $\left({}^{RL}I_{p,q}^0\mathsf{\omega }\right)\left(t\right)=\mathsf{\omega }\left(t\right)$, where ${(t-qs)}_{p,q}^{(\mathsf{\alpha }-1)}$ is a $(p,q)$-power function [27]. It is natural that, if $p=1$ and $q\to 1$, then the above $(p,q)$-integral ${}^{RL}I_{p,q}^{\mathsf{\alpha }}$ is reduced to the fractional Riemann–Liouville integral of order $\mathsf{\alpha }$ [1] given by

$$\left({}^{RL}I^{\mathsf{\alpha }}\mathsf{\omega }\right)\left(t\right)={\int }_0^t{\displaystyle \frac{{(t-s)}^{\mathsf{\alpha }-1}}{\Gamma \left(\mathsf{\alpha }\right)}}\mathsf{\omega }\left(s\right)\mathrm{d}s.$$

Also, with the same hypotheses, the fractional $(p,q)$-Caputo-type derivative ${}^{C}D_{p,q}^{\mathsf{\alpha }}$ is defined for $\mathsf{\omega }$ as

$$\left({}^{C}D_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\right)\left(t\right)=\left({}^{RL}I_{p,q}^{m-\mathsf{\alpha }}\left(D_{p,q}^m\mathsf{\omega }\right)\right)\left(t\right),m-1<\mathsf{\alpha }\le m,$$

where $D_{p,q}^m$ is the $(p,q)$-derivative of order $m\in \mathbb{N}$ [27]. It is notable that, if $p=1$ and $q\to 1$, then the fractional $(p,q)$-Caputo-type derivative ${}^{C}D_{p,q}^{\mathsf{\alpha }}$ is reduced to the fractional Caputo derivative ${}^{C}D_{p,q}^{\mathsf{\alpha }}$ given by

$$\left({}^{C}D^{\mathsf{\alpha }}\mathsf{\omega }\right)\left(t\right)={\int }_0^t{\displaystyle \frac{{(t-s)}^{m-\mathsf{\alpha }-1}}{\Gamma (m-\mathsf{\alpha })}}\left(D^{\left(m\right)}\mathsf{\omega }\right)\left(s\right)\mathrm{d}s,$$

where $D^{\left(m\right)}$ is the ordinary derivative of order m [28].

Note that, if $\mathsf{\omega }\left(t\right)=t^{\mathsf{\rho }},\mathsf{\rho }>-1$, then

• $\left({}^{RL}I_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\right)\left(t\right)={\displaystyle \frac{{\Gamma }_{p,q}(\mathsf{\rho }+1)}{{\Gamma }_{p,q}(\mathsf{\rho }+\mathsf{\alpha }+1)}}{t}^{\mathsf{\rho }+\mathsf{\alpha }},$
• $\left({}^{C}D_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\right)\left(t\right)={\displaystyle \frac{{\Gamma }_{p,q}(\mathsf{\rho }+1)}{{\Gamma }_{p,q}(\mathsf{\rho }-\mathsf{\alpha }+1)}}{t}^{\mathsf{\rho }-\mathsf{\alpha }}.$

Put $m=\left[\mathsf{\alpha }\right]+1$. Then,

$${}^{RL}I_{p,q}^{\mathsf{\alpha }}\left[{}^{C}D_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\left(t\right)\right]=\mathsf{\omega }\left(t\right)-\sum _{j=0}^{m-1}{\displaystyle \frac{D_{p,q}^j\mathsf{\omega }\left(0\right)}{{\Gamma }_{p,q}(j+1)p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}}}{t}^j.$$

The simplified version of the above equation is as follows:

$${}^{RL}I_{p,q}^{\mathsf{\alpha }}\left[{}^{C}D_{p,q}^{\mathsf{\alpha }}\mathsf{\omega }\left(t\right)\right]=\mathsf{\omega }\left(t\right)+C_0+C_{1}t+\dots +C_{m-1}{t}^{m-1},$$

where $C_j\in \mathbb{R};j=0,1,\dots ,m-1$ [27].

3. Main Results

The present section is going to state and prove the main existence theorems. Moreover, to establish the stability of $(p,q)$-solutions of the $(p,q)$-difference Langevin boundary problem (1), we extract the general $(p,q)$-integral form of $(p,q)$-solutions. The next lemma does this for us. Indeed, this lemma shall show that the $(p,q)$-difference Langevin boundary problem (1) has a $(p,q)$-integral solution as follows:

$$\begin{array}{cc}\hfill \mathsf{\omega }\left(t\right)& ={\displaystyle \frac{-\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\mathsf{\mu }{t}^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}g\left(p^{\mathsf{\beta }+1}s\right){\mathrm{d}}_{p,q}s,\hfill \end{array}$$

(4)

if we let $g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right):=g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))+g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))$.

Now, we prove the correctness of this claim in the follwoing lemma.

Lemma 1.

If we consider the hypotheses given in (2) and assume that $g\in C\left(\right[0,T],\mathbb{R})$, then $\mathsf{\omega }:[0,T]\to \mathbb{R}$ is a $(p,q)$-solution of the linear $(p,q)$-difference Langevin boundary problem (1) if and only if it satisfies the $(p,q)$-integral Equation (4).

Proof. 

Consider the equality $g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right):=g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))+g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))$. Then, the $(p,q)$-difference Langevin equation given in (1) is reduced to

$${}^{C}D_{p,q}^{\mathsf{\alpha }}\left({}^{C}D_{p,q}^{\mathsf{\beta }}+\mathsf{\mu }\right)\mathsf{\omega }\left(t\right)=g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right).$$

(5)

On both sides of the above $(p,q)$-difference equation, the fractional $(p,q)$-RL-integral of order $\mathsf{\alpha }$ is applied, and it yields

$$\left({}^{C}D_{p,q}^{\mathsf{\beta }}+\mathsf{\mu }\right)\mathsf{\omega }\left(t\right)=I_{p,q}^{\mathsf{\alpha }}g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)+a_0,$$

(6)

or, accordingly,

$$\begin{array}{cc}\hfill {}^{C}D_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(t\right)& =-\mathsf{\mu }\mathsf{\omega }\left(t\right)+I_{p,q}^{\mathsf{\alpha }}g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)+a_0\hfill \\ \hfill & =-\mathsf{\mu }\mathsf{\omega }\left(t\right)+{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }-1)}g\left(p^{\mathsf{\beta }+1}s\right){\mathrm{d}}_{p,q}s+a_0.\hfill \end{array}$$

(7)

On the other side, the fractional $(p,q)$-RL-integral of order $\mathsf{\beta }$ is used on both sides of (7), and it yields

$$\begin{array}{cc}\hfill \mathsf{\omega }\left(t\right)& =-\mathsf{\mu }{I}_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(t\right)+I_{p,q}^{\mathsf{\alpha }+\mathsf{\beta }}g\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)+a_0{\displaystyle \frac{t^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+a_1\hfill \\ \hfill & ={\displaystyle \frac{-\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +a_0{\displaystyle \frac{t^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+a_1.\hfill \end{array}$$

(8)

The anti-periodic condition $\mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left(pT\right)$ and (8) yield

$$\begin{array}{cc}\hfill a_1& ={\displaystyle \frac{\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -a_0{\displaystyle \frac{p^{\mathsf{\beta }}{T}^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}-a_1.\hfill \end{array}$$

(9)

Then,

$$\begin{array}{cc}\hfill a_0& ={\displaystyle \frac{\mathsf{\mu }\mathsf{\beta }}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{{\Gamma }_{p,q}(\mathsf{\beta }+1)}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s.\hfill \end{array}$$

(10)

Now, the periodic condition ${}^{C}D_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(0\right)={}^{C}D_{p,q}^{\mathsf{\beta }}\mathsf{\omega }\left(pT\right)$ and (7) provide

$$-2\mathsf{\mu }\mathsf{\omega }\left(0\right)={\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}g\left(p^{\mathsf{\beta }+1}s\right){\mathrm{d}}_{p,q}s.$$

Then,

$$\begin{array}{c}\hfill a_1=-{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}g\left(p^{\mathsf{\beta }+1}s\right){\mathrm{d}}_{p,q}s.\end{array}$$

(11)

The constants $a_0$ and $a_1$ obtained in (10) and (11), respectively, can be substituted into (8). By some simple computations, we get

$$\begin{array}{cc}\hfill \mathsf{\omega }\left(t\right)& ={\displaystyle \frac{-\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}\mathsf{\mu }}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}g\left(ps\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}g\left(p^{\mathsf{\beta }+1}s\right){\mathrm{d}}_{p,q}s.\hfill \end{array}$$

The converse is evident, and the proof is completed. □

In this step, we introduce a new Banach space that consists of all continuous functions like $\mathsf{\omega }:[0,1]\to \mathbb{R}$ under the sup-norm $\parallel \mathsf{\omega }\parallel =sup\{|\mathsf{\omega }\left(t\right)|,t\in [0,T\left]\right\}$. This space is denoted by the symbol $\mathbb{A}$. In other words, $(\mathbb{A}:=C([0,1],\mathbb{R}\left)\right),\parallel .\parallel )$ is a Banach space. Moreover, the above lemma helps us to define a new operator according to the given fractional $(p,q)$-Langevin problem (1). So, $G:\mathbb{A}\to \mathbb{A}$ is defined by

$$\begin{array}{cc}\hfill \left(G\mathsf{\omega }\right)& \left(t\right)={\displaystyle \frac{-\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}\mathsf{\mu }}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}[g_1(ps,\mathsf{\omega }\left(ps\right))+g_2(ps,\mathsf{\omega }\left(ps\right))]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}[g_1(ps,\mathsf{\omega }\left(ps\right))+g_2(ps,\mathsf{\omega }\left(ps\right))]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}[g_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))+g_2(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))]{\mathrm{d}}_{p,q}s.\hfill \end{array}$$

(12)

In the following, we consider some hypotheses that are needed for proving the main theorems of this paper. We enlist them for the continuous functions $g_1,g_2:[0,T]\times \mathbb{R}\to \mathbb{R}$ as follows:

• $\left(1C\right)$ There is a constant $A{g}_1>0$ so that the Lipschitz inequality

  $$|g_1(t,\mathsf{\omega })-g_1(t,\tilde{\mathsf{\omega }})|\le A{g}_1|\mathsf{\omega }-\tilde{\mathsf{\omega }}|$$

  holds for each $t\in [0,T]$ and $\mathsf{\omega },\tilde{\mathsf{\omega }}\in \mathbb{R}$;
• $\left(2C\right)$ There is a constant $A{g}_2>0$ so that the Lipschitz inequality

  $$|g_2(t,\mathsf{\omega })-g_2(t,\tilde{\mathsf{\omega }})|\le A{g}_2|\mathsf{\omega }-\tilde{\mathsf{\omega }}|$$

  holds for each $t\in [0,T]$ and $\mathsf{\omega },\tilde{\mathsf{\omega }}\in \mathbb{R}$;
• $\left(3C\right)$ There are two continuous functions $F_1,F_2:[0,T]\to {\mathbb{R}}^{+}$ so that, for all $(t,\mathsf{\omega })\in [0,T]\times \mathbb{R}$, $|g_1(t,\mathsf{\omega })|\le F_1\left(t\right)$ and $|g_2(t,\mathsf{\omega })|\le F_2\left(t\right)$, where, for $j=1,2,\parallel F_i\parallel =su{p}_{t\in [0,T]}|F_i\left(t\right)|$.

Along with the above conditions, Krasnoselskii’s fixed-point theorem [29] will be applied to prove the existence property for the given fractional $(p,q)$-Langevin problem (1). To be easy for the readers, we define some symbols as follows:

$$\begin{array}{cc}\hfill M_0& :={\displaystyle \frac{2|\mathsf{\mu }|T^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}},\hfill \\ \hfill M_1& :={\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}.\hfill \end{array}$$

(13)

Theorem 1.

Assume that $g_1,g_2:[0,T]\times \mathbb{A}\to \mathbb{A}$ are two continuous functions such that two conditions $\left(2C\right)$ and $\left(3C\right)$ hold. Let $B:=M_{1}A{g}_2<1$, where $M_1$ is introduced in (13). Then, there is at least one solution on $[0,T]$ for the fractional $(p,q)$-Langevin problem (1).

Proof. 

In view of the constants introduced in (13), we define $\{{\mathbb{B}}_{r_1}:=\mathsf{\omega }\in \mathbb{A}:\parallel \mathsf{\omega }\parallel \le r_1\}$ so that

$$r_1\ge {\displaystyle \frac{(\parallel F_1\parallel +\parallel F_2\parallel )M_1}{1-M_0}},M_0<1.$$

(14)

To complete the process of the proof, we need to establish all the conditions of Krasnoselskii’s fixed-point theorem. To accomplish this, two operators are defined on ${\mathbb{B}}_{r_1}$ as follows

$$\begin{array}{cc}\hfill \left(G_1\mathsf{\omega }\right)& \left(t\right)={\displaystyle \frac{-\mathsf{\mu }}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}\mathsf{\mu }}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{g}_1(ps,\mathsf{\omega }\left(ps\right)){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{g}_1(ps,\mathsf{\omega }\left(ps\right)){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}{g}_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right)){\mathrm{d}}_{p,q}s,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left(G_2\mathsf{\omega }\right)& \left(t\right)={\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{g}_2(ps,\mathsf{\omega }\left(ps\right)){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{g}_2(ps,\mathsf{\omega }\left(ps\right)){\mathrm{d}}_{p,q}s\hfill \\ \hfill & -{\displaystyle \frac{1}{2\mathsf{\mu }{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}{g}_2(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right)){\mathrm{d}}_{p,q}s.\hfill \end{array}$$

For each $\mathsf{\omega },\tilde{\mathsf{\omega }}\in {\mathbb{B}}_{r_1}$ and by $\left(3C\right)$, we have

$$\begin{array}{cc}\hfill |\left(G_1\mathsf{\omega }\right)\left(t\right)& +\left(G_2\tilde{\mathsf{\omega }}\right)\left(t\right)|\le {\displaystyle \frac{|\mathsf{\mu }|}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}|\mathsf{\mu }|}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_2(p^{\mathsf{\beta }+1}s,\tilde{\mathsf{\omega }}\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le {\displaystyle \frac{r_1{T}^{\mathsf{\beta }}|\mathsf{\mu }|}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+{\displaystyle \frac{r_1{T}^{\mathsf{\beta }}|\mathsf{\mu }|}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+{\displaystyle \frac{\parallel F_1\parallel T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{\parallel F_1\parallel p^{\mathsf{\alpha }}{T}^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}\hfill \\ \hfill & +{\displaystyle \frac{\parallel F_1{\parallel \left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}+{\displaystyle \frac{\parallel F_2\parallel T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{\parallel F_2\parallel p^{\mathsf{\alpha }}{T}^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{\parallel F_2{\parallel \left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\hfill \\ \hfill & \le r_1\left[{\displaystyle \frac{2|\mathsf{\mu }|T^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}\right]+\parallel F_1\parallel \left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]\hfill \\ \hfill & +\parallel F_2\parallel \left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]\hfill \\ \hfill & =r_1{M}_0+(\parallel F_1\parallel +\parallel F_2\parallel )M_1\le r_1.\hfill \end{array}$$

So, $\parallel G_1\mathsf{\omega }+G_2\tilde{\mathsf{\omega }}\parallel \le r_1$, and, thus, $G_1\mathsf{\omega }+G_2\tilde{\mathsf{\omega }}\in {\mathbb{B}}_{r_1}$; i.e., $(G_1+G_2)\left({\mathbb{B}}_{r_1}\right)\subseteq {\mathbb{B}}_{r_1}$.

In the following, since $g_1$ is a continuous function, $G_1$ has the property of the continuity. Now, the uniform boundedness of $G_1$ is investigated. By the condition $\left(3C\right)$ and the definition of ${\mathbb{B}}_{r_1}$, for every $\mathsf{\omega }\in {\mathbb{B}}_{r_1}$, we have

$$\begin{array}{cc}\hfill |\left(G_1\mathsf{\omega }\right)\left(t\right)|& \le {\displaystyle \frac{|\mathsf{\mu }|}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}\mathsf{\mu }}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le r_1\left[{\displaystyle \frac{2|\mathsf{\mu }|T^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}\right]+\parallel F_1\parallel \left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]<\infty .\hfill \end{array}$$

As the next step, we prove that $G_1$ is equicontinuous.

Let $t_1,t_2\in [0,T]$ with $t_1<t_2$. Then, for each $\mathsf{\omega }\in {\mathbb{B}}_{r_1}$, we have

$$\begin{array}{cc}\hfill |\left(G_1\mathsf{\omega }\right)\left(t_2\right)-\left(G_1\mathsf{\omega }\right)\left(t_1\right)|& \le {\displaystyle \frac{|\mathsf{\mu }|r_1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{t_1}\left[{(t_2-qs)}_{p,q}^{(\mathsf{\beta }-1)}-{(t_1-qs)}_{p,q}^{\mathsf{\beta }-1}\right]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{|\mathsf{\mu }|r_1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_{t_1}^{t_2}{(t_2-qs)}_{p,q}^{(\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & {\displaystyle \frac{(t_2^{\mathsf{\beta }}-t_1^{\mathsf{\beta }})|\mathsf{\mu }|r_1}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel F_1\parallel }{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{t_1}\left[{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}-{(t_1-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}\right]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel F_1\parallel }{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_{t_1}^{t_2}{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{(t_2^{\mathsf{\beta }}-t_1^{\mathsf{\beta }})\parallel F_1\parallel }{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s.\hfill \end{array}$$

(15)

Then, $|\left(G_1\mathsf{\omega }\right)\left(t_2\right)-\left(G_1\mathsf{\omega }\right)\left(t_1\right)|$ tends to zero because the right-hand side of (15) tends to zero when $t_2$ goes to $t_1$, and this convergence is independent of $\mathsf{\omega }\in {\mathbb{B}}_{r_1}$. The Arzela–Ascoli theorem provides the compactness of $G_1$ on ${\mathbb{B}}_{r_1}$. It is in turn demonstrated that $G_2$ is a contraction. For each $\mathsf{\omega },\tilde{\mathsf{\omega }}\in {\mathbb{B}}_{r_1}$, and, by the condition $\left(2C\right)$, we have

$$\begin{array}{cc}\hfill |\left(G_2\mathsf{\omega }\right)\left(t\right)& -\left(G_2\tilde{\mathsf{\omega }}\right)\left(t\right)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\mathsf{\omega }\left(ps\right))-g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\mathsf{\omega }\left(ps\right))-g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_2(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))-g_2(p^{\mathsf{\beta }+1}s,\tilde{\mathsf{\omega }}\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le {\displaystyle \frac{A{g}_2}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|\mathsf{\omega }\left(ps\right))-\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}A{g}_2}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|\mathsf{\omega }\left(ps\right))-\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{A{g}_2}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))-\tilde{\mathsf{\omega }}\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le \left[{\displaystyle \frac{T^{\mathsf{\alpha }+\mathsf{\beta }}A{g}_2}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{p^{\mathsf{\alpha }}{T}^{\mathsf{\alpha }+\mathsf{\beta }}A{g}_2}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}A{g}_2}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel \hfill \\ \hfill & =\left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]A{g}_2\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel \hfill \\ \hfill & =M_{1}A{g}_2\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel =B\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel .\hfill \end{array}$$

$G_2$ is a contraction since $B:=M_{1}A{g}_2<1$.

After the above steps, it is found that all assumptions given in Krasnoselskii’s fixed-point theorem are established. Hence, we apply its conclusion and obtain that there is a fixed point for the operator $G_1+G_2$; therefore, the fractional $(p,q)$-Langevin problem (1) has a solution on $[0,T]$ and the proof is completed. □

The hypotheses of the Leray–Schauder alternative theorem [29] help us to prove another existence theorem. At first, we provide two other conditions for this purpose.

• $\left(4C\right)$ There are two increasing functions ${\mathfrak{f}}_{\mathfrak{1}},{\mathfrak{f}}_{\mathfrak{2}}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and two continuous functions ${\mathfrak{h}}_{\mathfrak{1}},{\mathfrak{h}}_{\mathfrak{2}}:[0,1]\to {\mathbb{R}}^{+}$ such that

  $$\left\{\begin{array}{c}|g_1(t,w)|\le {\mathfrak{h}}_{\mathfrak{1}}\left(t\right){\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel ),\hfill \\ |g_2(t,w)|\le {\mathfrak{h}}_{\mathfrak{2}}\left(t\right){\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ),\hfill \end{array}\right.\forall (t,\mathsf{\omega })\in [0,T]\times \mathbb{R}.$$

  (16)
• $\left(5C\right)$ A constant $\delta >0$ exists such that

  $$\delta >{\displaystyle \frac{M_1\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}\left(\delta \right)+M_1\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}\left(\delta \right)}{1-M_0}},M_0\ne 1$$

  (17)

Theorem 2.

Assume that $g_1,g_2:[0,T]\times \mathbb{A}\to \mathbb{A}$ are two continuous functions such that two conditions $\left(4C\right)$ and $\left(5C\right)$ hold. Then, there is at least one solution on $[0,T]$ for the fractional $(p,q)$-Langevin problem (1).

Proof. 

To prove this theorem, we again consider the continuous operator $G:\mathbb{A}\to \mathbb{A}$ in (12). In the first step, it is claimed that G maps the bounded sets into the bounded sets in $\mathbb{A}$. To reach this goal, a bounded set is defined as ${\mathbb{B}}_{r_2}:=\{\mathsf{\omega }\in \mathbb{A}:\parallel \mathsf{\omega }\parallel \le r_2\}$ for $r_2>0$. In this case, the given inequalities (16) in the condition $\left(4C\right)$ imply that

$$\begin{array}{cc}\hfill |(G\mathsf{\omega })& \left(t\right)|\le {\displaystyle \frac{|\mathsf{\mu }|}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}|\mathsf{\mu }|}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|[g_1(ps,\mathsf{\omega }\left(ps\right))+g_2(ps,\mathsf{\omega }\left(ps\right))]|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|[g_1(ps,\mathsf{\omega }\left(ps\right))+g_2(ps,\mathsf{\omega }\left(ps\right))]|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|[g_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))+g_2(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))]|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le {\displaystyle \frac{|\mathsf{\mu }|}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}\parallel \mathsf{\omega }\parallel {\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}|\mathsf{\mu }|}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}\parallel \mathsf{\omega }\parallel {\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ){\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le {\displaystyle \frac{T^{\mathsf{\beta }}|\mathsf{\mu }|\parallel \mathsf{\omega }\parallel }{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+{\displaystyle \frac{T^{\mathsf{\beta }}|\mathsf{\mu }|\parallel \mathsf{\omega }\parallel }{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}+{\displaystyle \frac{T^{\mathsf{\alpha }+\mathsf{\beta }}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}\hfill \\ \hfill & +{\displaystyle \frac{p^{\mathsf{\alpha }}{T}^{\mathsf{\alpha }+\mathsf{\beta }}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\hfill \\ \hfill & +{\displaystyle \frac{T^{\mathsf{\alpha }+\mathsf{\beta }}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel )}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{p^{\mathsf{\alpha }}{T}^{\mathsf{\alpha }+\mathsf{\beta }}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel )}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel )}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\hfill \\ \hfill & =M_0+M_1\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )+M_2\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ),\hfill \end{array}$$

for each $\mathsf{\omega }\in {\mathbb{B}}_{r_2}$, and $\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel ={sup}_{[0,T]}|h_1\left(t\right)|$ and $\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel ={sup}_{[0,T]}|h_2\left(t\right)|$. So,

$$\parallel G\mathsf{\omega }\parallel \le M_0{r}_2+M_1\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}\left(r_1\right)+M_2\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}\left(r_2\right).$$

(18)

Inequality (18) shows the uniform boundedness of G on ${\mathbb{B}}_{r_2}$. Moreover, G maps every bounded set (like ${\mathbb{B}}_{r_2}$) into the equicontinuous set in $\mathbb{A}$. Indeed, if we take $\mathsf{\omega }\in {\mathbb{B}}_{r_2}$ and $t_1,t_2\in [0,T]$ with $t_1<t_2$, then the formulation (12) and the condition $\left(4C\right)$ yield

$$\begin{array}{cc}\hfill |\left(G\mathsf{\omega }\right)\left(t_2\right)& -\left(G\mathsf{\omega }\right)\left(t_1\right)|\le {\displaystyle \frac{|\mathsf{\mu }|r_2}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{t_1}[{(t_2-qs)}_{p,q}^{(\mathsf{\beta }-1)}-{(t_1-qs)}_{p,q}^{(\mathsf{\beta }-1)}]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{|\mathsf{\mu }|r_2}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_{t_1}^{t_2}{(t_2-qs)}_{p,q}^{(\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & {\displaystyle \frac{(t_2^{\mathsf{\beta }}-t_1^{\mathsf{\beta }})|\mathsf{\mu }|r_2}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right){\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}\left(r_2\right)}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{t_1}[{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}-{(t_1-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}\left(r_2\right)}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_{t_1}^{t_2}{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{(t_2^{\mathsf{\beta }}-t_1^{\mathsf{\beta }})\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}\left(r_2\right)}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right){\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}\left(r_2\right)}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{t_1}[{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}-{(t_1-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}]{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}\left(r_2\right)}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_{t_1}^{t_2}{(t_2-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{(t_2^{\mathsf{\beta }}-t_1^{\mathsf{\beta }})\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}\left(r_2\right)}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right){\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}{\mathrm{d}}_{p,q}s.\hfill \end{array}$$

(19)

By the right-hand side of (19), we determine that $|\left(G\mathsf{\omega }\right)\left(t_2\right)-\left(G\mathsf{\omega }\right)\left(t_1\right)|\to 0$ as $t_2-t_1\to 0$, independent of $\mathsf{\omega }\in {\mathbb{B}}_{r_2}$. Then, G is completely continuous by the Arzela–Ascoli theorem.

In the following, let us consider $\mathsf{\omega }$ as a solution of the equation $\mathsf{\omega }=cG\mathsf{\omega }$ for $c\in (0,1)$. If we repeat the proof of the uniform boundedness of G for each $t\in [0,T]$, then we have

$$\begin{array}{cc}\hfill |\mathsf{\omega }(t)|& =|c(G\mathsf{\omega }\left)\right(t)|\hfill \\ \hfill & \le |c||(G\mathsf{\omega }\left)\right(t)|\hfill \\ \hfill & \le M_0\parallel \mathsf{\omega }\parallel +M_1\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )+M_2\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel ).\hfill \end{array}$$

Hence,

$$\parallel \mathsf{\omega }\parallel \le {\displaystyle \frac{M_1\parallel {\mathfrak{h}}_{\mathfrak{1}}\parallel {\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )+M_1\parallel {\mathfrak{h}}_{\mathfrak{2}}\parallel {\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel )}{1-M_0}}.$$

Now, the given inequality (17) in the condition $\left(5C\right)$ states that a constant like $\delta$ exists so that $\parallel \mathsf{\omega }\parallel \ne \delta$. Define $U_{\delta }=\{\mathsf{\omega }\in \mathbb{A}:\parallel \mathsf{\omega }\parallel <\delta \}$.

Therefore, it is evident that $G:{\overline{U}}_{\delta }\to \mathbb{A}$ is both continuous and completely continuous. Such a choice regarding $U_{\delta }$ ensures that there is no $\mathsf{\omega }\in \partial U_{\delta }$ such that $\mathsf{\omega }=cG\mathsf{\omega }$ for some $c\in (0,1)$. On the basis of the Leray–Schauder-type theorem, it thus follows that G has a fixed point in ${\overline{U}}_{\delta }$.

Therefore, a solution exists for the fractional $(p,q)$-Langevin problem (1) and the proof is completed. □

Along with the above conditions $\left(1C\right)$ and $\left(2C\right)$ and by the existing hypotheses in the Banach contraction principle [29], we can prove the uniqueness for the solutions.

Theorem 3.

Assume that $g_1,g_2:[0,T]\times \mathbb{A}\to \mathbb{A}$ are two continuous functions such that two conditions $\left(1C\right)$ and $\left(2C\right)$ hold. Then, there is a unique solution for the fractional $(p,q)$-Langevin problem (1) if

$$A_G:=M_0+M_{1}A{g}_1+M_{1}A{g}_2<1,$$

is a contraction constant for the operator G defined by (12). Here, the constants $M_0$ and $M_1$ are introduced in (13).

Proof. 

On the basis of the Banach contraction principle, it is sufficient that we prove the uniqueness of fixed point for $G:\mathbb{A}\times \mathbb{A}$, which is corresponding to the uniqueness of solutions of the $(p,q)$-Langevin problem (1). By defining

$$\underset{t\in [0,T]}{sup}|g_1(t,0)|:=g_1^{*}<\infty ,\underset{t\in [0,T]}{sup}|g_2(t,0)|:=g_2^{*}<\infty ,$$

and choosing

$$r_2\ge {\displaystyle \frac{M_1{g}_1^{*}+M_1{g}_2^{*}}{1-(M_0+M_{1}A{g}_1+M_{1}A{g}_2)}},$$

we show that the inclusion relation $G{\mathbb{B}}_{r_2}\subset {\mathbb{B}}_{r_2}$ holds so that

$${\mathbb{B}}_{r_2}:=\{\mathsf{\omega }\in \mathbb{A}:\parallel \mathsf{\omega }\parallel \le r_2\}.$$

If we continue the arguments from the proof of Theorem 1 for each $\mathsf{\omega }\in {\mathbb{B}}_{r_2}$, then we get

$$\parallel G\mathsf{\omega }\parallel \le (M_0+M_{1}A{g}_1+M_{1}A{g}_2)r_2+M_1{g}_1^{*}+M_1{g}_2^{*}<r_2,$$

and this yields $G\mathsf{\omega }\in {\mathbb{B}}_{r_2}$ immediately. In other words, G maps ${\mathbb{B}}_{r_2}$ into itself. Finally, investigation of the contractive behavior for G is the last step. Let $\mathsf{\omega },\tilde{\mathsf{\omega }}\in \mathbb{A}$ be arbitrary. Then,

$$\begin{array}{cc}\hfill & |\left(G\mathsf{\omega }\right)\left(t\right)-\left(G\tilde{\mathsf{\omega }}\right)\left(t\right)|\le {\displaystyle \frac{|\mathsf{\mu }|}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)-\tilde{\mathsf{\omega }}\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}|\mathsf{\mu }|}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\beta }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\beta }-1)}|\mathsf{\omega }\left(p^{1-\mathsf{\beta }}s\right)-\tilde{\mathsf{\omega }}\left(p^{1-\mathsf{\beta }}s\right)|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))-g_1(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_1(ps,\mathsf{\omega }\left(ps\right))-g_1(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_1(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))-g_1(p^{\mathsf{\beta }+1}s,\tilde{\mathsf{\omega }}\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^t{(t-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\mathsf{\omega }\left(ps\right))-g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{t^{\mathsf{\beta }}}{{\left(pT\right)}^{\mathsf{\beta }}{p}^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }+\mathsf{\beta }}{2}\right)}{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta })}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }+\mathsf{\beta }-1)}|g_2(ps,\mathsf{\omega }\left(ps\right))-g_2(ps,\tilde{\mathsf{\omega }}\left(ps\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & +{\displaystyle \frac{1}{2|\mathsf{\mu }|p^{\left(\genfrac{}{}{0pt}{}{\mathsf{\alpha }}{2}\right)}{\Gamma }_{p,q}\left(\mathsf{\alpha }\right)}}{\int }_0^{pT}{(pT-qs)}_{p,q}^{(\mathsf{\alpha }-1)}|g_2(p^{\mathsf{\beta }+1}s,\mathsf{\omega }\left(p^{\mathsf{\beta }+1}s\right))-g_2(p^{\mathsf{\beta }+1}s,\tilde{\mathsf{\omega }}\left(p^{\mathsf{\beta }+1}s\right))|{\mathrm{d}}_{p,q}s\hfill \\ \hfill & \le \left[{\displaystyle \frac{2|\mathsf{\mu }|T^{\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\beta }+1)}}\right]\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel +\left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]A{g}_1\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel \hfill \\ \hfill & +\left[{\displaystyle \frac{(1+p^{\mathsf{\alpha }})T^{\mathsf{\alpha }+\mathsf{\beta }}}{{\Gamma }_{p,q}(\mathsf{\alpha }+\mathsf{\beta }+1)}}+{\displaystyle \frac{{\left(pT\right)}^{\mathsf{\alpha }}}{2|\mathsf{\mu }|{\Gamma }_{p,q}(\mathsf{\alpha }+1)}}\right]A{g}_2\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel \hfill \\ \hfill & =(M_0+M_{1}A{g}_1+M_{1}A{g}_2)\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel \hfill \\ \hfill & =A_G\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel .\hfill \end{array}$$

(20)

According to the obtained results in (20), it follows that G is a contraction; i.e.,

$$\parallel G\mathsf{\omega }-G\tilde{\mathsf{\omega }}\parallel \le A_G\parallel \mathsf{\omega }-\tilde{\mathsf{\omega }}\parallel ;A_G<1.$$

As we had claimed, G has a unique fixed point in $\mathbb{A}$. Therefore, the fractional $(p,q)$-Langevin equation has a unique solution in the Banach space $\mathbb{A}=\left(C\right([0,1],\mathbb{R}),\parallel .\parallel )$. □

4. Examples

Some illustrative examples are designed in this section for validation of our results. Note that these examples are different from the structural point of view, but they are the same from the data point of view because the main difference in our results is related to the structure of the functions.

A. Illustrative example for Theorem 1:

As the first example, we consider a $(p,q)$-difference Langevin problem (taken from the main problem (1)) given by

$$\left\{\begin{array}{c}{}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{5}}}\left({}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}+{\displaystyle \frac{1}{8}}\right)\mathsf{\omega }\left(t\right)={\displaystyle \frac{{0.75}^{\frac{3}{5}}t}{9}}\left({\displaystyle \frac{|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}{1+|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}}+sin\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)\right),\hfill \\ \mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),\hfill \\ {}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left(0\right)={}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),t\in [0,1].\hfill \end{array}\right.$$

(21)

These parameters are used in the above $(p,q)$-difference problem:

$$p={\displaystyle \frac{3}{4}},q={\displaystyle \frac{1}{2}},\mathsf{\alpha }={\displaystyle \frac{2}{5}},\mathsf{\beta }={\displaystyle \frac{1}{5}},\mathsf{\mu }={\displaystyle \frac{1}{8}},T=1.$$

(22)

Moreover, we define two continuous functions

$$\left\{\begin{array}{c}{g}_1\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)\right)={\displaystyle \frac{0.841t}{9}}\left({\displaystyle \frac{|\mathsf{\omega }(0.841t)|}{1+|\mathsf{\omega }(0.841t)|}}\right),\hfill \\ g_2\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)\right)={\displaystyle \frac{0.841t}{9}}sin\mathsf{\omega }\left(0.841t\right).\hfill \end{array}\right.$$

Clearly, $g_1,g_2\in C([0,1]\times \mathbb{R},\mathbb{R})$. For establishing the conclusion of Theorem 1, we need to show that both conditions $\left(2C\right)$ and $\left(3C\right)$ are satisfied. For this purpose, for each $\mathsf{\omega },\tilde{\mathsf{\omega }}\in \mathbb{A}:=C([0,1],\mathbb{R})$, we have

$$\begin{array}{c}\hfill |g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)\right)-g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\tilde{\mathsf{\omega }}\left(\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right)\right)|\\ \hfill \le {\displaystyle \frac{0.841t}{9}}|sin\mathsf{\omega }\left(0.841t\right)-sin\tilde{\mathsf{\omega }}\left(0.841t\right)|\\ \hfill \le {\displaystyle \frac{0.841t}{9}}|\mathsf{\omega }\left(0.841t\right)-\tilde{\mathsf{\omega }}\left(0.841t\right)|\\ \hfill \le {\displaystyle \frac{0.841}{9}}|\mathsf{\omega }\left(0.841t\right)-\tilde{\mathsf{\omega }}\left(0.841t\right)|.\end{array}$$

The above Lipschitz condition provides the Lipschitz constant $A_{g_2}={\displaystyle \frac{0.841}{9}}>0$. This confirms the fulfillment of the condition $\left(2C\right)$. On the other hand, we have (for each $\mathsf{\omega }\in \mathbb{A}$)

$$\left\{\begin{array}{c}|g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))|=|{\displaystyle \frac{0.841t}{9}}\left({\displaystyle \frac{|\mathsf{\omega }(0.841t)|}{1+|\mathsf{\omega }(0.841t)|}}\right)|\le {\displaystyle \frac{0.841t}{9}},\hfill \\ |g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))|=|{\displaystyle \frac{0.841t}{9}}sin\mathsf{\omega }\left(0.841t\right)|\le {\displaystyle \frac{0.841t}{9}}.\hfill \end{array}\right.$$

That is, $F_1\left(t\right)=F_2\left(t\right)={\displaystyle \frac{0.841t}{9}}$ and $\parallel F_i\parallel ={sup}_{[0,1]}|F_i\left(t\right)|={\displaystyle \frac{0.841}{9}}\simeq 0.0934$ for $i=1,2$. This confirms the fulfillment of the condition $\left(3C\right)$. Moreover, by considering the given parameters in (22), we get $M_0=0.236$, $M_1=4.76707$, and, accordingly, we obtain

$$B:=M_{1}A{g}_2\simeq 0.445244392<1.$$

All assumptions in Theorem 1 are satisfied for the $(p,q)$-difference Langevin problem (21). Therefore, this problem has at least one solution by Theorem 1.

B. Illustrative example for Theorem 2: As the second example, we consider a $(p,q)$-difference Langevin problem (taken from the main problem (1)) given by

$$\left\{\begin{array}{c}{}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{5}}}\left({}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}+{\displaystyle \frac{1}{8}}\right)\mathsf{\omega }\left(t\right)={\displaystyle \frac{{0.75}^{\frac{3}{5}}{t}^2}{120}}\left({\displaystyle \frac{1}{2}}cos\mathsf{\omega }({0.75}^{{\displaystyle \frac{3}{5}}}t)+{\displaystyle \frac{|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}{3+|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}}\right)\hfill \\ +{\displaystyle \frac{0.841}{440+t}}\left({\displaystyle \frac{4}{5}}+|tan\mathsf{\omega }({0.75}^{{\displaystyle \frac{3}{5}}}t)|\right),\hfill \\ \mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),\hfill \\ {}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left(0\right)={}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),t\in [0,1].\hfill \end{array}\right.$$

(23)

We use the same parameters given in (22) and define

$$\left\{\begin{array}{c}{g}_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))={\displaystyle \frac{{0.75}^{\frac{3}{5}}{t}^2}{120}}\left({\displaystyle \frac{1}{2}}cos\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)+{\displaystyle \frac{|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}{3+|\mathsf{\omega }({0.75}^{\frac{3}{5}}t)|}}\right),\hfill \\ g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))={\displaystyle \frac{{0.75}^{\frac{3}{5}}}{440+t}}\left({\displaystyle \frac{4}{5}}+|tan\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)|\right).\hfill \end{array}\right.$$

For establishing the conclusion of Theorem 2, we need to show that both conditions $\left(4C\right)$ and $\left(5C\right)$ are satisfied. For each $(t,\mathsf{\omega })\in [0,1]\times \mathbb{A}$, we have

$$\left\{\begin{array}{c}|g_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))|={\displaystyle \frac{0.841t^t}{120}}(1+\parallel \mathsf{\omega }\parallel ),\hfill \\ |g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))|={\displaystyle \frac{0.841}{440+t}}(1+\parallel \mathsf{\omega }\parallel ).\hfill \end{array}\right.$$

Based on the above inequalities, we have ${\mathfrak{h}}_{\mathfrak{1}}\left(\mathfrak{t}\right)={\displaystyle \frac{0.841t^2}{120}}$ and ${\mathfrak{h}}_{\mathfrak{2}}\left(\mathfrak{t}\right)={\displaystyle \frac{0.841}{440+t}}$. Also, ${\mathfrak{f}}_{\mathfrak{1}}(\parallel \mathsf{\omega }\parallel )={\mathfrak{f}}_{\mathfrak{2}}(\parallel \mathsf{\omega }\parallel )=\parallel \mathsf{\omega }\parallel +1$. Therefore, $\parallel \mathfrak{h}\mathfrak{1}\parallel =0.007008$, $\parallel \mathfrak{h}\mathfrak{2}\parallel =0.001911$, and ${\mathfrak{f}}_{\mathfrak{1}}\left(\delta \right)={\mathfrak{f}}_{\mathfrak{2}}\left(\delta \right)=\delta +1$.

Again, by applying the parameters given in (22), we get $M_0=0.236<1$ and $M_1=4.76707$. Therefore, the inequality (17) in the condition $\left(5C\right)$ is satisfied for $\delta >0.05892854$. Then, the existence of at least one solution is confirmed for the fractional $(p,q)$-Langevin problem (23) by the conclusion of Theorem 2.

C. Illustrative example for Theorem 3:

As the third example, we consider a $(p,q)$-difference Langevin problem (taken from the main problem (1)) given by

$$\left\{\begin{array}{c}{}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{5}}}\left({}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}+{\displaystyle \frac{1}{8}}\right)\mathsf{\omega }\left(t\right)={\displaystyle \frac{1}{759+{0.75}^{\frac{3}{5}}{e}^t}}\left(|sin\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)|+|tan\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)|\right),\hfill \\ \mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),\hfill \\ {}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left(0\right)={}^{C}D_{{\displaystyle \frac{3}{4}},{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{5}}}\mathsf{\omega }\left({\displaystyle \frac{3}{4}}\right),t\in [0,1].\hfill \end{array}\right.$$

(24)

With the parameters defined above (in (22)), we define

$$\left\{\begin{array}{c}{g}_1(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))={\displaystyle \frac{1}{759+{0.75}^{\frac{3}{5}}{e}^t}}|sin\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)|,\hfill \\ g_2(p^{\mathsf{\alpha }+\mathsf{\beta }}t,\mathsf{\omega }\left(p^{\mathsf{\alpha }+\mathsf{\beta }}t\right))={\displaystyle \frac{1}{759+{0.75}^{\frac{3}{5}}{e}^t}}|tan\mathsf{\omega }\left({0.75}^{{\displaystyle \frac{3}{5}}}t\right)|.\hfill \end{array}\right.$$

Clearly, the Lipschitz inequalities are satisfied for $g_1$ and $g_2$ so that $A{g}_1\simeq A{g}_2\simeq 0.001316>0$; i.e., the conditions $\left(1C\right)$ and $\left(2C\right)$ are valid. On the other hand, since $M_0\simeq 0.236$ and $M_1\simeq 4.76707$,

$$A_G:=M_0+M_{1}A{g}_1+M_{1}A{g}_2\simeq 0.248546<1.$$

Theorem 3 follows that the fractional $(p,q)$-Langevin problem (24) has a unique solution.

5. Conclusions

This paper investigated another form of the well-known Langevin equation in the framework of a two-parameter $(p,q)$-derivative pattern. This type of Langevin equation depends on two parameters, $0<q<p\le 1$, which comprise the basis of the theory of post-quantum calculus. We obtained a $(p,q)$-integral equation as an equivalent $(p,q)$-solution of the given sequential $(p,q)$-difference Langevin boundary value problems (1) and (2). We studied the existence property for such solutions on the basis of a fixed-point-based approach. In fact, we applied the Krasnoselskii and Leray–Schauder fixed-point theorems for proving the existence results. Also, the Banach contraction principle was used to finalize our results regarding the uniqueness property. Numerical examples illustrated the validity of the theorems. It is natural that, if we put $p=1$, then we obtain the q-version of this two-term sequential equation as follows:

$$\left\{\begin{array}{c}{}^{C}D_q^{\mathsf{\alpha }}\left({}^{C}D_q^{\mathsf{\beta }}+\mathsf{\mu }\right)\mathsf{\omega }\left(t\right)=g_1(t,\mathsf{\omega }\left(t\right))+g_2(t,\mathsf{\omega }\left(t\right)),\hfill \\ \mathsf{\omega }\left(0\right)=-\mathsf{\omega }\left(T\right),{}^{C}D_q^{\mathsf{\beta }}\mathsf{\omega }\left(0\right)={}^{C}D_q^{\mathsf{\beta }}\mathsf{\omega }\left(T\right).\hfill \end{array}\right.$$

If $q\to 1$, then the Caputo fractional Langevin equation is obtained immediately. For future studies, we aim to extend some numerical algorithms to find the approximate $(p,q)$-solutions because there is no numerical algorithm for approximating solutions of the nonlinear $(p,q)$-difference boundary problems.