Existence of Solutions for Caputo-Type Fractional (p,q)-Difference Equations Under Robin Boundary Conditions

Abstract

In this paper, we investigate the existence results of solutions for Caputo-type fractional $(p,q)$-difference equations. Using Banach’s fixed-point theorem, we obtain the existence and uniqueness results. Meanwhile, by applying Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, we also obtain the existence results of non-trivial solutions. Finally, we provide examples to verify the correctness of the given results. Moreover, relevant applications are presented through specific examples.

1. Introduction

In this paper, we investigate the $(p,q)$-difference equations under Robin boundary conditions, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }x(t)=f(p^{\alpha }t,x(p^{\alpha }t)),t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\hfill \\ a_{1}x(0)+b_1{\mathcal{D}}_{p,q}x(0)=c_1,\hfill \\ a_{2}x({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})+b_2{\mathcal{D}}_{p,q}x({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})=c_2,\hfill \end{array}\right.\end{array}\end{array}$$

(1)

where

$f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\times \mathbb{R},\mathbb{R})$${, }^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }$ and ${\mathcal{D}}_{p,q}$ respectively denote the Caputo-type fractional $(p,q)$-derivative operator and the first-order $(p,q)$-difference operator, $\alpha \in \left(1,2\right]$, $0<q<p\le 1$, $\mathcal{T}>0$, and $a_i,b_i,c_i(i=1,2)$ are real constants.

Fractional calculus can be used to study non-integer order derivatives and integrals, providing new tools and methods for solving practical problems in real-world applications. Fractional calculus has become an indispensable tool in modern science and engineering, with applications spanning fluid flow, electrical networks, rheology, and many other fields [1,2]. Its ability to model complex phenomena with greater accuracy and depth has opened up new possibilities for research and development in these areas. More detailed introductions to fractional calculus can be found in [3].

Quantum calculus, also known as q-calculus, represents a part of generalized calculus. Initially, Jackson [4,5] defined the operational properties of q-calculus. With the rise of quantum mechanics, quantum calculus emerged based on the need for a tool to describe the characteristics of quantum systems. Quantum calculus replaces derivatives with difference operators, which provides research ideas for studying some non-differentiable equations. Kac and Cheung [6] studied some properties of fractional q-derivatives and integrals. Al-Salam [7] provided a detailed introduction to q-calculus and presented its applications in physics, including theories such as quantum groups, quantum algebras, and string theory. For more detailed information, readers can refer to [8]. Li et al. [9] studied the positive solutions of a class of Riemann–Liouville-type q-difference equations through the classical fixed-point theorem on cones. Allouch et al. [10] carried out a discussion on the existence results of solutions for a class of q-difference equations under nonlinear integral boundary conditions. Relying on the features of q-integrals and q-derivatives, Tariboon et al. [11] put forward a q-shift operator and added to the idea of fractional quantum calculus. Recently, many scholars have conducted further research on q-calculus and achieved new results. Readers can refer to [12,13,14,15] and references therein.

Chakrabarti [16] further developed quantum calculus by generalizing q-calculus to $(p,q)$-calculus. Compared with q-calculus, $(p,q)$-calculus differs in terms of definitions, theorems, and operational properties. Under certain conditions, $(p,q)$-calculus can also be regarded as a variation of q-calculus. In particular, we have found that the $(p,q)$-integral degenerates into the q-integral when $p=1$. Inspired by the applications of q-calculus, many scholars have conducted in-depth research on $(p,q)$-calculus, and $(p,q)$-calculus theory has been effectively applied in fields such as number theory, combinatorics, physical sciences, and more [17,18,19,20].

Soontharanon and Sitthiwirattham [19] respectively studied the concepts and operational properties of Caputo-type and Riemann–Liouville-type $(p,q)$-difference operators. Sadjang [21] introduced the polynomial basis for the $(p,q)$-derivative, investigated its corresponding properties, and put forward the $(p,q)$-Taylor formula. Roh et al. [22] conducted an in-depth investigation into the properties of $(p,q)$-calculus, specifically focusing on $(p,q)$-exponential functions and their inverses. Their research was primarily driven by the analysis of solutions to both linear and nonlinear $(p,q)$-difference equations, which provided valuable insights into the behavior and characteristics of these functions within the $(p,q)$-framework. In a separate study, Neang et al. [23] extended the exploration of $(p,q)$-calculus by examining inequalities within a finite interval. Their work not only contributed to the theoretical understanding of $(p,q)$-calculus but also established significant results in the form of trapezoidal and midpoint-type inequalities for fractional $(p,q)$-calculus. These findings carry significant implications for the application of $(p,q)$-calculus across diverse mathematical contexts, particularly in areas involving fractional calculus and its extensions. The contributions from both sets of researchers have significantly advanced the field of $(p,q)$-calculus, providing a solid foundation for future studies and applications in related areas. Some classical fixed-point theorems, such as the Banach fixed-point theorem, Schaefer’s fixed-point theorems, Leray–Schauder alternative, Mönch’s fixed-point theorem, and so on, have been widely applied in the study of the existence of solutions to differential equations. Recently, these fixed-point theorem tools have been widely applied. For example, Ahmad et al. [24] utilized fixed-point data to obtain the existence and uniqueness of fixed-point solutions within the framework of a class of chaotic models and controlled metric spaces; Dimitrov et al. [25] utilized the Brouwer fixed-point theorem and the Banach fixed-point theorem to obtain the existence and uniqueness of solutions for a class of Nabla fractional differential equations to produce generalized Ulam–Hyers–Rassias-stable solutions; and Srivastava et al. [26] discussed the existence, uniqueness, and multiplicity of solutions for a class of Riemann–Liouville difference equations defined under the $\delta$ fractional version by using the fixed-point theorem. It is worth noting that classical fixed-point theorems have been proven to be effective in dealing with the existence of solutions to $(p,q)$-difference equations. Readers can refer to [20,27,28,29,30,31,32,33,34,35,36,37,38,39,40].

In [27], Qin and Sun studied fractional $(p,q)$-difference equations, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathcal{R}}{\mathcal{D}}_{p,q}^{\alpha }u(t)+f(p^{\alpha }t,u(p^{\alpha }t))=0,t\in (0,1),\hfill \\ u(0)=u(1)=0,\hfill \end{array}\right.\end{array}\end{array}$$

(2)

where $f\in \mathcal{C}([0,1]\times \mathbb{R},\mathbb{R})$, $\alpha \in \left(1,2\right]$, $0<q<p\le 1$, and ${}^{\mathcal{R}}{\mathcal{D}}_{p,q}^{\alpha }$ represents the Riemann–Liouville-type fractional $(p,q)$-difference operator. The existence conditions of solutions to problem (2) were obtained by applying the fixed-point theorem and the generalized Banach contraction principle. Moreover, the sufficient conditions for the existence of at least one solution were separately established. In [28], Nuntigrangjana et al. obtained the existence and uniqueness results of solutions for first-order and second-order $(p,q)$-difference equations through the Banach contraction mapping principle. This achievement not only enriches the theoretical understanding of these equations but also provides practical guidance for related applications. Meanwhile, Liu and Liu [29] studied the existence and multiplicity of positive solutions for a class of $(p,q)$-difference equations based on the Krein–Rutman theorem and topological degree theory. Their findings have significant implications for the study of positive solution related problems and open up new directions for further research in this field.

In [20], Agarwal et al. investigated a class of $(p,q)$-difference equations, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }u(t)=f(t,u(p^{\alpha }t)),t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\hfill \\ u(0)+u({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})=0,{\mathcal{D}}_{p,q}u(0)+{\mathcal{D}}_{p,q}u({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})=0,\hfill \end{array}\right.\end{array}\end{array}$$

(3)

where $f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\times \mathbb{R},\mathbb{R})$, $\mathcal{T}>0$, $1<\alpha \le 2$, ${}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }$ and ${\mathcal{D}}_{p,q}$ represent the Caputo-type fractional $(p,q)$-difference operator and the first-order $(p,q)$-difference operator respectively. The basis for the existence of solutions to Problem (3) was provided, and supplementary research on $(p,q)$-difference equations under first-order anti-periodic boundary conditions was carried out by applying the fixed-point theorem.

In [34], Promsakon et al. considered the following second-order $(p,q)$-difference equations under separated boundary conditions:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{D}}_{p,q}^{2}u(t)=f(t,u(p^{2}t)),t\in \left[0,{\displaystyle \frac{\mathcal{T}}{p^2}}\right],\hfill \\ a_{1}u(0)+b_1{\mathcal{D}}_{p,q}u(0)=c_1,\hfill \\ a_{2}u(\mathcal{T})+b_2{\mathcal{D}}_{p,q}u\left({\displaystyle \frac{\mathcal{T}}{p}}\right)=c_2,\hfill \end{array}\right.\end{array}\end{array}$$

(4)

where $f\in \mathcal{C}([0,{\displaystyle \frac{T}{p^2}}]\times \mathbb{R},\mathbb{R})$, $0<q<p\le 1$, ${\mathcal{D}}_{p,q}^2$ is the second-order $(p,q)$-difference operator, $\mathcal{T}>0$ is a constant, and $a_i,b_i,c_i(i=1,2)$ are also constants. The existence conditions of (4) for solutions were derived using the tools of classical fixed-point theorems.

In [35], Mesmouli et al. discussed the following fractional $(p,q)$-difference equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{}^{\mathcal{R}}{\mathcal{D}}_{p,q}^{\alpha }u(t)+\psi (p^{\alpha }t,u(p^{\alpha }t))={\mathcal{D}}_{p,q}^{\alpha -1}h(pt,u(pt)),t\in (0,1),\hfill \\ u(0)=0,u(1)={\int }_0^{1}h(pr,u(pr))d_{p,q}r,\hfill \end{array}\right.\end{array}\end{array}$$

(5)

where $\psi ,h:[0,1]\times E\to E$, E is a Banach space with a norm $\parallel ·\parallel$, $\alpha \in (1,2]$, $0<q<p\le 1$, and ${}^{\mathcal{R}}{\mathcal{D}}_{p,q}^{\alpha }$ is the Riemann–Liouville-type fractional $(p,q)$-derivative operator. They obtained the existence results of solutions for Problem (5) by applying non-compactness and Mönch’s fixed-point theorem.

Motivated by the above papers [20,27,34,35], we discuss the existence results of non-trivial solutions for Caputo-type fractional $(p,q)$-difference equations under Robin boundary conditions (1). Our work extends previous research through two key innovations. First, we analyze a distinctive nonlinear term $f(p^{\alpha }t,x(p^{\alpha }t))$ where the temporal scaling factor $p^{\alpha }$ introduces delayed feedback effects. This structure fundamentally differs from the conventional formulations in [20,22], where nonlinear terms depend directly on t rather than on its scaled counterpart $p^{\alpha }t$. Next, when $a_i,b_i,c_i$ ($i=1,2$) take specific values, it can be concluded that Problem (1) serves as a further generalization of the boundary conditions mentioned above. For example, when $a_1=a_2=0$, the boundary conditions in (1) are Neumann boundary conditions, while when $b_1=b_2=0$, the boundary conditions in (1) are Dirichlet-type boundary conditions [27]. Robin-type boundary conditions play an important role in dealing with electromagnetic problems, heat transfer problems, and diffusion equations in the fields of chemistry and biology [41]. Meanwhile, we present an application which bridges fractional difference equations and theoretical physics.

The rest of this paper is structured as follows: in Section 2, we present some definitions and lemmas about q-calculus and $(p,q)$-calculus; in Section 3, we establish the conditions for the existence of solutions to Problem (1) by applying the techniques associated with fixed-point theorems; in Section 4, we present three examples to verify our results; in Section 5, real-world applications of Problem (1) in this article are presented; finally, Section 6 presents the conclusions of this paper.

2. Preliminaries

To begin with, we revisit several fundamental concepts related to q-calculus and $(p,q)$-calculus. For a more comprehensive and in-depth introduction to these topics, readers can refer to [7,8,11,19,21,42,43,44].

Let $[\mathfrak{a},\mathfrak{b}]\subset \mathbb{R}$, and $0<q<p\le 1$. We define

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left[\vartheta \right]}_{p,q}={\displaystyle \frac{p^{\vartheta }-q^{\vartheta }}{p-q}},\vartheta \in \mathbb{N},\hfill \\ & {\left[\vartheta \right]}_{p,q}!={\left[\vartheta \right]}_{p,q}{\left[\vartheta -1\right]}_{p,q}\cdots {\left[1\right]}_{p,q}=\prod _{i=1}^{\vartheta }{\displaystyle \frac{p^i-q^i}{p-q}},\vartheta \in \mathbb{N},{\left[0\right]}_{p,q}!=1.\hfill \end{array}\end{array}$$

Definition 1.

Choose $\vartheta \in {\mathbb{N}}_0$, ${\mathbb{N}}_0=\left\{0,1,2,\cdots \right\}$, $\mathfrak{a},\mathfrak{b}\in \mathbb{R}$. Then, the definition of the power function ${(\mathfrak{a}-\mathfrak{b})}_q^{(\vartheta )}$ and ${(\mathfrak{a}-\mathfrak{b})}_{p,q}^{(\vartheta )}$ is as follows:

$${(\mathfrak{a}-\mathfrak{b})}_q^{(n)}=\prod _{\vartheta =0}^{n-1}(\mathfrak{a}-\mathfrak{b}{q}^{\vartheta }),{(\mathfrak{a}-\mathfrak{b})}_{p,q}^{(n)}=\prod _{\vartheta =0}^{n-1}(\mathfrak{a}{p}^{\vartheta }-\mathfrak{b}{q}^{\vartheta }).$$

Specifically,

$${(\mathfrak{a}-\mathfrak{b})}_q^{(0)}={(\mathfrak{a}-\mathfrak{b})}_{p,q}^{(0)}=1$$

when $n=0$.

Remark 1.

It is notable that ${(\mathfrak{a}-\mathfrak{b})}_{p,q}^{(n)}$ can be approximated by ${(\mathfrak{a}-\mathfrak{b})}_q^{(n)}$ when $p=1$, thereby establishing a connection between the functions ${(\mathfrak{a}-\mathfrak{b})}_q^{(n)}$ and ${(\mathfrak{a}-\mathfrak{b})}_{p,q}^{(n)}$.

Definition 2.

For any $t\in \mathbb{R}\setminus \left\{0,-1,-2,\cdots \right\}$, the $(p,q)$-gamma function is defined as follows:

$${\Gamma }_{p,q}(t)={\displaystyle \frac{{(p-q)}_{p,q}^{(t-1)}}{{(p-q)}^{t-1}}}.$$

This is equivalent to

$${\Gamma }_{p,q}(t)=p^{{\displaystyle \frac{t(t-1)}{2}}}{\int }_0^{\infty }{x}^{t-1}{E}_{p,q}^{-qx}{d}_{p,q}x,$$

where

$$E_{p,q}^{-qx}=\sum _{\vartheta =0}^{\infty }{\displaystyle \frac{q^{\left(\genfrac{}{}{0pt}{}{\vartheta }{2}\right)}}{{\left[\vartheta \right]}_{p,q}!}}{(-qx)}^{\vartheta },\left(\begin{array}{c}\vartheta \\ 2\end{array}\right)=\frac{\Gamma (\vartheta +1)}{\Gamma (2+1)\Gamma (\vartheta -2+1)}.$$

It is easy to deduced that ${\Gamma }_{p,q}(t+1)={\left[t\right]}_{p,q}{\Gamma }_{p,q}(t)$.

Subsequently, we introduce some definitions of q-derivatives, q-integrals, $(p,q)$-derivatives, and $(p,q)$-integrals.

Definition 3

([7]). Let f be a real valued function. Then, the q-derivative of f is defined by

$${\mathcal{D}}_{q}f(t)={\displaystyle \frac{f(t)-f(qt)}{t-qt}},t\ne 0.$$

It is notable that ${\displaystyle {\mathcal{D}}_{q}f(0)=\underset{t\to 0}{lim}{\mathcal{D}}_{q}f(t)}$.

Definition 4

([7]). The q-integral of a function f defined in the interval $[0,\mathcal{T}]$ is defined by

$${\int }_0^{t}f(s)d_{q}s=(1-q)t\sum _{n=0}^{\infty }{q}^{n}f\left(q^{n}t\right),t\in [0,\mathcal{T}].$$

Definition 5

([21]). Let $f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\to \mathbb{R})$ and differentiable at 0. Then, the $(p,q)$-derivative of f is defined by

$${\mathcal{D}}_{p,q}f(t)={\displaystyle \frac{f(pt)-f(qt)}{pt-qt}},t\ne 0,p\ne q,$$

provided that f is differentiable at 0. Furthermore, ${\displaystyle {\mathcal{D}}_{p,q}f(0)=\underset{t\to 0}{lim}{\mathcal{D}}_{p,q}f(t)}$ holds.

Definition 6

([21]). Let $f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\to \mathbb{R})$. Then, the $(p,q)$-integral of f is defined by

$${\int }_0^{t}f(s)d_{p,q}s=(p-q)t\sum _{n=0}^{\infty }{\displaystyle \frac{q^n}{p^{n+1}}}f\left({\displaystyle \frac{q^n}{p^{n+1}}}t\right)$$

whenever the series on the right-hand side converges.

Definition 7

([19]). Let $f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\to \mathbb{R})$. Then, for $\mathcal{T}>0$ and $\alpha \ge 0$, the Caputo-type fractional $(p,q)$-difference operator is defined by

$$({\mathcal{D}}_{p,q}^{\alpha }f)(t)=\left({}^{\mathcal{C}}{\mathcal{I}}_{p,q}^{\omega -\alpha }{\mathcal{D}}_{p,q}^{\omega }f\right)(t)$$

and ${(}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{0}f)(t)=f(t)$, where ω is the smallest integer greater than or equal to α.

Lemma 1

([19]). Let f be continuous. Then, for any $\alpha ,\beta \ge 0$, the $(p,q)$-integral and $(p,q)$-derivative satisfy the following operational rules:

(i) 

$({\mathcal{I}}_{p,q}^{\beta }{\mathcal{I}}_{p,q}^{\alpha }f)(t)=({\mathcal{I}}_{p,q}^{\alpha +\beta }f)(t)$;

(ii) 

$({\mathcal{D}}_{p,q}^{\alpha }{\mathcal{I}}_{p,q}^{\alpha }f)(t)=f(t)$;

(iii) 

${\displaystyle ({\mathcal{I}}_{p,q}^{\alpha }{\mathcal{D}}_{p,q}^{n}f)(t)=({\mathcal{D}}_{p,q}^n{\mathcal{I}}_{p,q}^{\alpha }f)(t)-\sum _{\vartheta =0}^{[\alpha ]-1}\frac{t^{\alpha -n+\vartheta }}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha -n+\vartheta +1)}({\mathcal{D}}_{p,q}^{\vartheta }f)(0),n\in \mathbb{N}}$;

(iv) 

${\displaystyle ({{\mathcal{I}}_{p,q}^{\alpha }}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }f)(t)=f(t)-\sum _{\vartheta =0}^{[\alpha ]-1}\frac{t^{\vartheta }}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\vartheta +1)}({\mathcal{D}}_{p,q}^{\vartheta }f)(0)}$.

In the following, we put forward a lemma to define the solution of Problem (1).

Lemma 2.

Let $a_i,b_i,c_i(i=1,2)$ be constants. Suppose that $\mathfrak{h}\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R})$; then, the unique solution of the boundary value problem

$$\left\{\begin{array}{c}{}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{\alpha }x(t)=\mathfrak{h}(t),t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\hfill \\ a_{1}x(0)+b_1{\mathcal{D}}_{p,q}x(0)=c_1,\hfill \\ a_{2}x({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})+b_2{\mathcal{D}}_{p,q}x({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})=c_2,\hfill \end{array}\right.$$

(6)

is provided by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x(t)={\Lambda }_{1}t+{\Lambda }_2& +{\displaystyle \frac{(b_1-a_{1}t)a_2}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -1}}}\right)d_{p,q}r\hfill \\ & +{\displaystyle \frac{(b_1-a_{1}t)b_2}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -2}}}\right)d_{p,q}r\hfill \\ & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -1}}}\right)d_{p,q}r,\hfill \end{array}\end{array}$$

where ${\Lambda }_1,{\Lambda }_2,\Lambda$ are defined as follows:

$$\begin{array}{c}\hfill {\Lambda }_1={\displaystyle \frac{a_1{c}_2-c_1{a}_2}{\Lambda }},{\Lambda }_2={\displaystyle \frac{c_1(a_2(\frac{\mathcal{T}}{p^{\alpha }})+b_2-b_1{c}_2)}{\Lambda }},\Lambda =a_1(a_2({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})+b_2-b_1{a}_2)\ne 0.\end{array}$$

Proof. 

First, we integrate on both sides of the equation in (6) at the same time; then, we have

$$x(t)={\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -1}}}\right)d_{p,q}r+d_{1}t+d_2,$$

(7)

where $t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$ and where $d_1,d_2$ are two constants. Because the boundary conditions in (6) contain the first-order $(p,q)$-derivative, when we take the $(p,q)$-derivative of (7) we obtain

$$D_{p,q}x(t)={\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^t{(t-qr)}^{(\alpha -2)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -2}}}\right)d_{p,q}r+d_1.$$

(8)

According to the boundary conditions in (6), $d_1$ and $d_2$ can be deduced through Equations (7) and (8). After simple calculations, we have

$$\begin{array}{c}\hfill d_1={\Lambda }_1-{\displaystyle \frac{1}{\Lambda }}\left[{\displaystyle \frac{a_1{a}_2}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -1}}}\right)d_{p,q}r\right.\\ \hfill \left.+{\displaystyle \frac{a_1{b}_2}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -2}}}\right)d_{p,q}r\right]\end{array}$$

and

$$\begin{array}{c}\hfill d_2={\Lambda }_2+{\displaystyle \frac{1}{\Lambda }}\left[{\displaystyle \frac{b_1{a}_2}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -1}}}\right)d_{p,q}r\right.\\ \hfill \left.+{\displaystyle \frac{b_1{b}_2}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\mathfrak{h}\left({\displaystyle \frac{r}{p^{\alpha -2}}}\right)d_{p,q}r\right].\end{array}$$

Substituting $d_1$ and $d_2$ into Equation (7), we obtain the solution for Problem (6). □

We define the space $E=\mathcal{C}\left([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R}\right)$ as the Banach space of all continuous functions from $\left[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}\right]$ to $\mathbb{R}$, with the norm of $\parallel x\parallel$ provided by $\parallel x\parallel =sup\left\{\left|x(t)\right|:t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\right\}$. Subsequently, we introduce the operator $\mathcal{A}:E\to E$ and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill (\mathcal{A}x)(t)={\Lambda }_{1}t+{\Lambda }_2& +{\displaystyle \frac{(b_1-a_{1}t)a_2}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}f(p^{\alpha -1}r,x(p^{\alpha -1}r))d_{p,q}r\hfill \\ & +{\displaystyle \frac{(b_1-a_{1}t)b_2}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}f(p^{\alpha }r,x(p^{\alpha }r))d_{p,q}r\hfill \\ & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}f(p^{\alpha -1}r,x(p^{\alpha -1}r))d_{p,q}r.\hfill \end{array}\end{array}$$

(9)

It is worth noting that if the operator equation $\mathcal{A}x=x$ has a fixed-point, then Problem (1) has a solution. Consequently, we can convert the problem of finding the solution of the equation into that of finding the fixed point of the equation.

3. Main Results

In this section, we mainly focus on the fixed-point theorem in order to explore the existence of solutions to the $(p,q)$-difference equation. We perform appropriate mathematical transformations on the $(p,q)$-difference equation to establish the corresponding mapping relationship. Finally, we use the theories in the fixed-point theorem to determine whether the constructed mapping has fixed points, and thereby determine whether the $(p,q)$-difference equation has solutions so as to obtain our expected results.

First, we employ the Banach fixed-point theorem to prove the existence and uniqueness of the solution for Problem (1). To simplify the calculation, we represent ${\Omega }_1$ and ${\Omega }_2$ in the following way:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\Omega }_1=(\left|b_1\right|+\left|a_1\right|({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}))\left|a_2\right|,{\Omega }_2=(\left|b_1\right|+\left|a_1\right|({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}))\left|b_2\right|.\end{array}\end{array}$$

Theorem 1.

Assume that $f\in \mathcal{C}(([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R})\to \mathbb{R})$ and satisfies the following assumptions:

• $(H_1)$ Assume that $\mathcal{L}(x)$ is a $(p,q)$-integrable function with $\mathcal{L}(x):[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\to \mathbb{R}$. For any $x,y\in \mathbb{R}$, suppose $\left|f(t,x)-f(t,y)\right|\le \mathcal{L}(t)\left|x-y\right|$.

If $\Omega <1$, then Problem (1) has a unique solution on $[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$, where Ω is defined as

$$\begin{array}{c}\hfill \begin{array}{cc}& \Omega ={\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r\hfill \\ & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\right)d_{p,q}r\hfill \\ & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r.\hfill \end{array}\end{array}$$

(10)

Proof. 

We convert Problem (1) into a fixed-point problem of the form $x=\mathcal{A}x$. Subsequently, by invoking the Banach contraction principle, we aim to demonstrate that the operator $\mathcal{A}$ defined by (9) possesses a unique fixed point. We can construct a bounded ball $B_{\sigma }$ which is defined as $B_{\sigma }=\left\{x\in E:\parallel x\parallel \le \sigma \right\}$, where

$$\sigma \ge \left[\left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\mathcal{M}\Phi \right]{(1-\Omega )}^{-1}$$

and ${\displaystyle \mathcal{M}=\underset{t\in [0,\frac{\mathcal{T}}{p^{\alpha }}]}{sup}\left|f(t,0)\right|}$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \Phi ={\displaystyle \frac{{\Omega }_1{T}^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{{\Omega }_2{T}^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}+{\displaystyle \frac{T^{\alpha }}{p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}.\end{array}\end{array}$$

(11)

From the assumption $(H_1)$, we can infer that

$$\left|f(t,x(t))\right|\le \left|f(t,x(t))-f(t,0)\right|+\left|f(t,0)\right|\le \mathcal{L}(t)\sigma +\mathcal{M}.$$

(12)

Next, we prove that $\mathcal{A}{B}_{\sigma }\subset B_{\sigma }$. For any $x\in B_{\sigma }$, from (9) and (12) we have

$$\begin{array}{cc}\hfill & \left|(\mathcal{A}x)(t)\right|\hfill \\ \hfill & \le \left|{\Lambda }_{1}t\right|+\left|{\Lambda }_2\right|+{\displaystyle \frac{\left|b_1-a_{1}t\right|\left|a_2\right|}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{\left|b_1-a_{1}t\right|\left|b_2\right|}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left|f(p^{\alpha }r,x(p^{\alpha }r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ \hfill & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))-f(p^{\alpha -1}r,0)\right|+\left|f(p^{\alpha -1}r,0)\right|)d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}(\left|f(p^{\alpha }r,x(p^{\alpha }r))-f(p^{\alpha }r,0)\right|+\left|f(p^{\alpha }r,0)\right|)d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))-f(p^{\alpha -1}r,0)\right|+\left|f(p^{\alpha -1}r,0)\right|)d_{p,q}r\hfill \\ \hfill & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel x\parallel +\mathcal{M}\right)d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\parallel x\parallel +\mathcal{M}\right)d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel x\parallel +\mathcal{M}\right)d_{p,q}r\hfill \\ \hfill & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +\sigma \left\{{\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r\right.\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\right)d_{p,q}r\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r\right\}\hfill \\ \hfill & +\mathcal{M}\left\{{\displaystyle \frac{{\Omega }_1{T}^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{{\Omega }_2{T}^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}+{\displaystyle \frac{T^{\alpha }}{p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}\right\}.\hfill \end{array}$$

From (10) and (11), we have

$$\parallel \mathcal{A}x\parallel \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\sigma \Omega +\mathcal{M}\Phi \le \sigma .$$

This means that $\mathcal{A}{B}_{\sigma }\subset B_{\sigma }$. Then, for any $x,y\in E$, we have

$$\begin{array}{cc}\hfill & \parallel \mathcal{A}x-\mathcal{A}y\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))-f(p^{\alpha -1}r,y(p^{\alpha -1}r))\right|)d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}(\left|f(p^{\alpha }r,x(p^{\alpha }r))-f(p^{\alpha }r,y(p^{\alpha }r))\right|)|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))-f(p^{\alpha -1}r,y(p^{\alpha -1}r))\right|)d_{p,q}r\hfill \\ \hfill & \le {\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel x\parallel -\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel y\parallel )d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\parallel x\parallel -\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\parallel y\parallel )|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel x\parallel -\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\parallel y\parallel )d_{p,q}r\hfill \\ \hfill & \le \parallel x-y\parallel \left\{{\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r\right.\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha }}})\right)d_{p,q}r\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left(\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})\right)d_{p,q}r\right\},\hfill \end{array}$$

and from (10), we can obtain

$$\parallel \mathcal{A}x-\mathcal{A}y\parallel \le \Omega \parallel x-y\parallel .$$

It can be seen from condition $(H_1)$ that $\Omega \in (0,1)$; therefore, $\mathcal{A}$ is a contraction mapping. According to the Banach contraction mapping principle, Problem (1) has a unique solution on $[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$. □

Specifically, if $\mathcal{L}(t)$ is a constant function that is $\mathcal{L}(t)=\mathcal{L}$, we obtain

$$\parallel \mathcal{A}x-\mathcal{A}y\parallel \le \mathcal{L}\Phi \parallel x-y\parallel .$$

(13)

If $\mathcal{L}\in (0,{\displaystyle \frac{1}{\Phi }})$, then it can also be derived that $\mathcal{A}$ is a contraction mapping. Therefore, we have the following result.

Corollary 1.

Assume that the conditions in Theorem 1 are satisfied. If $\mathcal{L}(t)$ is a constant function defined by $\mathcal{L}$ using the inequality (13), then Theorem 1 is still valid.

Subsequently, we can utilize Krasnoselskii’s fixed-point theorem to demonstrate the existence of a non-trivial solution for Problem (1).

Lemma 3

([45]). Let M be a non-empty, closed, convex, and bounded subset of a Banach space E. Let ${\mathcal{A}}_1,{\mathcal{A}}_2$ be two operators such that:

(1) 

For any $x,y\in M$, ${\mathcal{A}}_{1}x+{\mathcal{A}}_{2}y\in M$.

(2) 

${\mathcal{A}}_1$ is a compact and continuous mapping.

(3) 

${\mathcal{A}}_2$ is a contraction mapping.

Then, there exists $z\in M$ such that $z={\mathcal{A}}_{1}z+{\mathcal{A}}_{2}z$.

Theorem 2.

Assume that $f\in \mathcal{C}(([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R})\to \mathbb{R})$ and satisfies condition $(H_1)$. In addition, the following assumptions still hold for function f:

$(H_2)$

$\left|f(t,x)\right|\le k(t)\psi (\parallel x\parallel ),(t,x)\in ([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\times \mathbb{R})$, $k,\psi \in C([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]\times \mathbb{R})$, and ψ is a non-decreasing function.

If

$${\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})d_{p,q}r<1,$$

(14)

then there exists at least one non-trivial solution of Problem (1) on $\left[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}\right]$.

Proof. 

First, we define the norm of k as follows:

$$\parallel k\parallel =\underset{t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]}{sup}\left|k(t)\right|$$

and choose

$$\tilde{\sigma }\ge \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\psi (\tilde{\sigma })\parallel k\parallel \Phi .$$

(15)

Considering the operator ${\mathcal{A}}_1,{\mathcal{A}}_2$ on $B_{\tilde{\sigma }}=\left\{x\in E:\parallel x\parallel \le \tilde{\sigma }\right\}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{A}}_{1}x(t)& ={\displaystyle \frac{(b_1-a_{1}t)}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{a}_2{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}f(p^{\alpha -1}r,x(p^{\alpha -1}r))d_{p,q}r\hfill \\ & +{\displaystyle \frac{(b_1-a_{1}t)b_2}{\Lambda p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}f(p^{\alpha }r,x(p^{\alpha }r))d_{p,q}r\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{A}}_{2}y(t)={\Lambda }_{1}t+{\Lambda }_2+{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}f(p^{\alpha -1}r,y(p^{\alpha -1}r))d_{p,q}r.\end{array}\end{array}$$

Then, can be noticed that ${\mathcal{A}}_{1}x+{\mathcal{A}}_{2}y=\mathcal{A}x$. Therefore, for any $x\in B_{\tilde{\sigma }}$, from (11) and (15) we obtain

$$\begin{array}{cc}\hfill & \parallel {\mathcal{A}}_{1}x(t)+{\mathcal{A}}_{2}x(t)\parallel \hfill \\ & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+{\displaystyle \frac{{\Omega }_1}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}k(p^{\alpha -1}t)\psi (\parallel x\parallel )d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{{\Omega }_2}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}k(p^{\alpha }t)\psi (\parallel x\parallel )d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}k(p^{\alpha -1}t)\psi (\parallel x\parallel )d_{p,q}r\hfill \\ \hfill & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +\psi (\tilde{\sigma })\parallel k\parallel \left\{{\displaystyle \frac{{\Omega }_1{\mathcal{T}}^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{{\Omega }_2{\mathcal{T}}^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}+{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}\right\}\hfill \\ \hfill & =\left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\psi (\tilde{\sigma })\parallel k\parallel \Phi \hfill \\ \hfill & \le \tilde{\sigma }.\hfill \end{array}$$

Consequently, from $(H_1)$ in Lemma 3, we have ${\mathcal{A}}_{1}x+{\mathcal{A}}_{2}y\in B_{\tilde{\sigma }}$. Therefore, item (1) of Lemma 3 is satisfied. Next, we prove that ${\mathcal{A}}_2$ is a contraction mapping. For any $x,y\in B_{\tilde{\sigma }}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|{\mathcal{A}}_{2}x(t)-{\mathcal{A}}_{2}y(t)\right|\hfill \\ & \le {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))-f(p^{\alpha -1}r,y(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ & \le {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})d_{p,q}r\parallel x-y\parallel ,\hfill \end{array}\end{array}$$

and from (14), we obtain $\left|{\mathcal{A}}_{2}x(t)-{\mathcal{A}}_{2}y(t)\right|\le \parallel x-y\parallel$, satisfying item (3) of Lemma 3. For $x\in B_{\tilde{\sigma }}$, we obtain

$$\parallel {\mathcal{A}}_{1}x\parallel \le \psi (r)\parallel k\parallel \left\{{\displaystyle \frac{{\Omega }_1{T}^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{{\Omega }_2{T}^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}\right\},$$

which implies that the operator ${\mathcal{A}}_1$ is uniformly bounded on $B_{\tilde{\sigma }}$. Subsequently, we proceed to demonstrate that the operator ${\mathcal{A}}_1$ is equicontinuous. Let ${\displaystyle \tilde{f}=\underset{t\in [0,\frac{\mathcal{T}}{p^{\alpha }}]\times B_{\tilde{\sigma }}}{sup}\left|f(t,x)\right|}$, where $t_1,t_2\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$ and $t_1<t_2$. For any $x\in B_{\tilde{\sigma }}$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|{\mathcal{A}}_{1}x(t_2)-{\mathcal{A}}_{1}x(t_1)\right|& \le {\displaystyle \frac{\left|a_1{a}_2\right|(t_2-t_1)}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ & +{\displaystyle \frac{\left|a_1{b}_2\right|(t_2-t_1)}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left|f(p^{\alpha }r,x(p^{\alpha }r))\right|d_{p,q}r\hfill \\ & \le \tilde{f}\left\{{\displaystyle \frac{\left|a_1{a}_2\right|T^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{\left|a_1{b}_2\right|T^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}\right\}(t_2-t_1)\hfill \end{array}\end{array}$$

when $t_1\to t_2$, $\left|{\mathcal{A}}_{1}x(t_2)-{\mathcal{A}}_{1}x(t_1)\right|\to 0$, which is independent of x. Consequently, ${\mathcal{A}}_1$ is equicontinuous. According to the Arzelà–Ascoli theorem, the operator ${\mathcal{A}}_1$ is compact within $B_{\tilde{\sigma }}$. Therefore, item (2) of Lemma 3 is satisfied. Hence, Problem (1) has at least one non-trivial solution on $[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$. □

In particular, assume that $\psi (\parallel x\parallel )=1$ and that the conditions of (15) are still satisfied. Obviously, the following result still holds.

Corollary 2.

Assume that the conditions in Theorem 2 are satisfied. If $\psi (\parallel x\parallel )=1$, then Theorem 2 still holds.

In the final part, we draw on Leray-Schauder’s nonlinear alternative to explore the existence of non-trivial solutions for Problem (1).

Lemma 4

([46]). Let E denote a Banach space and let C be a closed and convex subset of E. Now, consider U, which is an open subset of C and satisfies the condition that $0\in U$. Assume that $\mathcal{A}:\tilde{U}\to E$ is a continuous and compact mapping operator; then, $\mathcal{A}(\tilde{U})$ is a relatively compact subset of C. Now, one of the following two conditions is satisfied:

(1) 

The operator $\mathcal{A}$ has a fixed point in $\tilde{U}$.

(2) 

There exists $x\in \partial U$ ($\tilde{U}\subseteq E$) and $x=\lambda \mathcal{A}(x)$ when $\lambda \in (0,1)$.

Theorem 3.

Assume that $f\in \mathcal{C}([0,T/p^2]\times \mathbb{R}\to \mathbb{R})$ and satisfies the following conditions: $(H_3)$ Assume that $\mathcal{F}:[0,\infty )\to [0,\infty )$ is a continuous and non-decreasing function and let $\zeta \in C([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R})$; then, for any $(t,x)\in ([0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}],\mathbb{R})$, the following inequality is satisfied:

$$\left|f(t,x)\right|\le \zeta (t)\mathcal{F}(\parallel x\parallel ).$$

(16)

$(H_4)$ Choose $\mu >0$ as a constant, and

$${\displaystyle \frac{\mu }{\left|{\Lambda }_1(\frac{\mathcal{T}}{p^{\alpha }})\right|+\left|{\Lambda }_2\right|+\mathcal{F}(\mu )\parallel \zeta \parallel \Phi }}>1;$$

then, there exists at least one non-trivial solution of Problem (1) on $\left[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}\right]$.

Proof. 

Consider the operator $\mathcal{A}:E\to E$ defined by (9). We now prove that the operator $\mathcal{A}$ maps bounded sets in the domain $[0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$ to bounded sets. Let $\rho >0$ be a positive constant and define the set $B_{\rho }=\{x\in E:\parallel x\parallel \le \rho \}$. Then, for any $t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$. Making use of Equations (10) and (16), we can derive the following results:

$$\begin{array}{cc}\hfill \left|\mathcal{A}x(t)\right|& \le \left|{\Lambda }_{1}t\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +{\displaystyle \frac{\left|b_1-a_{1}t\right|\left|a_2\right|}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{\left|b_1-a_{1}t\right|\left|b_2\right|}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left|f(p^{\alpha }r,x(p^{\alpha }r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^t{(t-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ \hfill & \le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|\hfill \\ \hfill & +\mathcal{F}(\parallel x\parallel )\parallel \zeta \parallel \left[{\displaystyle \frac{{\Omega }_1{\mathcal{T}}^{\alpha }}{\left|\Lambda \right|p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}+{\displaystyle \frac{{\Omega }_2{\mathcal{T}}^{\alpha -1}}{\left|\Lambda \right|p^{\alpha }{p}^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha )}}+{\displaystyle \frac{{\mathcal{T}}^{\alpha }}{p^{{\alpha }^2}{p}^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha +1)}}\right]\hfill \\ \hfill & =\left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\mathcal{F}(\rho )\parallel \zeta \parallel \Phi ,\hfill \end{array}$$

and consequently we have

$$\left|\mathcal{A}x(t)\right|\le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\mathcal{F}(\rho )\parallel \zeta \parallel \Phi .$$

In the following, we demonstrate that the operator $\mathcal{A}$ is equicontinuous on E. Let $x\in B_{\rho }$ and $t_1<t_2$; then, for any $t_1,t_2\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$, we have

$$\begin{array}{cc}\hfill \left|\mathcal{A}x(t_2)-\mathcal{A}x(t_1)\right|& \le {\displaystyle \frac{\left|a_1{a}_2\right|(t_2-t_1)}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,x(p^{\alpha -1}r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{\left|a_1{b}_2\right|(t_2-t_1)}{\left|\Lambda \right|p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\Gamma }_{p,q}(\alpha -1)}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -2)}\left|f(p^{\alpha }r,x(p^{\alpha }r))\right|d_{p,q}r\hfill \\ \hfill & +{\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}\left[{\int }_0^{t_2}{(t_2-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,y(p^{\alpha -1}r))\right|d_{p,q}r\right.\hfill \\ \hfill & \left.-{\int }_0^{t_1}{(t_1-qr)}^{(\alpha -1)}\left|f(p^{\alpha -1}r,y(p^{\alpha -1}r))\right|d_{p,q}r\right].\hfill \end{array}$$

Evidently, the right-hand side of the inequality converges to 0 as $t_2\to t_1$ and this convergence is independent of the selection of $x\in B_{\rho }$. From the Arzelà–Ascoli theorem, it follows that the operator $\mathcal{A}:E\to E$ is completely continuous.

Therefore, after the boundedness of the set of all solutions to the equation $x=\lambda \mathcal{A}x$ with $\lambda \in (0,1)$ has been demonstrated, the outcome is obtained from the Leray-Schauder nonlinear alternative (Lemma 4). Next, assume that $x(t)$ is a solution for Problem (1). From condition $(H_3)$, for any $x\in [0,{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}]$ we have

$$\left|x(t)\right|\le \left|{\Lambda }_1({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}})\right|+\left|{\Lambda }_2\right|+\mathcal{F}(\parallel x\parallel )\parallel \zeta \parallel \Phi .$$

Thus, we obtain

$${\displaystyle \frac{\parallel x\parallel }{\left|{\Lambda }_1(\frac{\mathcal{T}}{p^{\alpha }})\right|+\left|{\Lambda }_2\right|+\mathcal{F}(\parallel x\parallel )\parallel \zeta \parallel \Phi }}\le 1.$$

In view of $(H_4)$, suppose that $\parallel x\parallel \ne \mu$. Defining the set U as

$$U=\left\{x\in E:\parallel x\parallel <\mu \right\},$$

it can be seen that $\mathcal{A}:\tilde{U}\to E$ is completely continuous. For the set U defined as above, there does not exist any $x\in \partial U$ such that $x=\lambda \mathcal{A}x$ holds when $\lambda \in (0,1)$. This contradicts (2) in Lemma 4. Hence, it can be known by applying Lemma 4 that there is at least one fixed point for $x\in \tilde{U}$, which also means that Problem (1) has at least one non-trivial solution. □

4. Examples

Three examples are presented in this section to further verify our results.

Example 1.

Let $f\in \mathcal{C}([0,2\sqrt{2}]\times \mathbb{R}\to \mathbb{R})$. We study the following $(p,q)$-difference equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{3/2}x(t)=\frac{\left|cos(\frac{t}{2\sqrt{2}})\right|}{20}\left(\frac{x^2(\frac{t}{2\sqrt{2}})+2|x(\frac{t}{2\sqrt{2}}|)}{\left|x(\frac{t}{2\sqrt{2}})\right|+1}\right)+e^{\frac{t}{2\sqrt{2}}},t\in [0,2\sqrt{2}],}\hfill \\ {\displaystyle \frac{3}{4}x(0)+\frac{3}{5}{\mathcal{D}}_{p,q}x(0)=\frac{1}{2},}\hfill \\ {\displaystyle \frac{1}{6}x(2\sqrt{2})+\frac{3}{4}{\mathcal{D}}_{p,q}x(2\sqrt{2})=\frac{1}{3}.}\hfill \end{array}\right.\end{array}\end{array}$$

(17)

Choose $\alpha ={\displaystyle \frac{3}{2}}$, $p={\displaystyle \frac{1}{2}}$, $q={\displaystyle \frac{1}{4}}$, $\mathcal{T}=1$, $a_1={\displaystyle \frac{3}{4}}$, $b_1={\displaystyle \frac{3}{5}}$, $c_1={\displaystyle \frac{1}{2}}$, $a_2={\displaystyle \frac{1}{6}}$, $b_2={\displaystyle \frac{3}{4}}$, $c_2={\displaystyle \frac{1}{3}}$. After simple calculations, we have ${\Lambda }_1\approx 0.3256$, ${\Lambda }_2\approx 0.5039$, $\Lambda \approx 0.5758$, ${\Omega }_1\approx 0.1884$, ${\Omega }_2\approx 0.8477$, $\Phi \approx 6.6032$, and $f(t,x)={\displaystyle \frac{\left|cost\right|}{20}}\left({\displaystyle \frac{x^2+2\left|x\right|)}{\left|x\right|+1}}\right)+e^t$. For any $t\in [0,2\sqrt{2}]$, we can conclude that

$$\left|f(t,x)-f(t,y)\right|\le {\displaystyle \frac{1}{10}}\left|x-y\right|$$

and

$$\mathcal{L}\Phi \approx 0.66032<1.$$

Therefore, by Theorem 1, Equation (17) has a unique solution on $t\in [0,2\sqrt{2}]$.

Example 2.

Let $f\in \mathcal{C}([0,2\sqrt{2}]\times \mathbb{R}\to \mathbb{R})$. We study the following $(p,q)$-difference equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{3/2}x(t)=\frac{1}{8}\left[\frac{e^{-\frac{t}{2\sqrt{2}}}}{2+{(\frac{t}{2\sqrt{2}})}^2}·\frac{x(\frac{t}{2\sqrt{2}})}{x(\frac{t}{2\sqrt{2}})+1})+\frac{t}{2\sqrt{2}}\right],t\in [0,2\sqrt{2}],}\hfill \\ {\displaystyle \frac{3}{4}x(0)+\frac{3}{5}{\mathcal{D}}_{p,q}x(0)=\frac{1}{2},}\hfill \\ {\displaystyle \frac{1}{6}x(2\sqrt{2})+\frac{3}{4}{\mathcal{D}}_{p,q}x(2\sqrt{2})=\frac{1}{3}.}\hfill \end{array}\right.\end{array}\end{array}$$

(18)

Choose $\alpha ={\displaystyle \frac{3}{2}}$, $p={\displaystyle \frac{1}{2}}$, $q={\displaystyle \frac{1}{4}}$, $\mathcal{T}=1$, $a_1={\displaystyle \frac{3}{4}}$, $b_1={\displaystyle \frac{3}{5}}$, $c_1={\displaystyle \frac{1}{2}}$, $a_2={\displaystyle \frac{1}{6}}$, $b_2={\displaystyle \frac{3}{4}}$, $c_2={\displaystyle \frac{1}{3}}$. After simple calculations, we have ${\Lambda }_1\approx 0.3256$, ${\Lambda }_2\approx 0.5039$, $\Lambda \approx 0.5758$, ${\Omega }_1\approx 0.1884$, ${\Omega }_2\approx 0.8477$, $\Phi \approx 6.6032$ and

$$\left|f({\displaystyle \frac{t}{2\sqrt{2}}},x({\displaystyle \frac{t}{2\sqrt{2}}}))\right|\le {\displaystyle \frac{1}{8}}\left[{\displaystyle \frac{e^{-\frac{t}{2\sqrt{2}}}}{2+{(\frac{t}{2\sqrt{2}})}^2}}+{\displaystyle \frac{t}{2\sqrt{2}}}\right]=k(t),t\in [0,2\sqrt{2}].$$

Based on the provided data, we can obtain $\mathcal{L}={\displaystyle \frac{1}{16}}$. For $\mathcal{L}={\displaystyle \frac{1}{16}}$, we are able to derive that

$$\left|f({\displaystyle \frac{t}{2\sqrt{2}}},x({\displaystyle \frac{t}{2\sqrt{2}}}))-f({\displaystyle \frac{t}{2\sqrt{2}}},y({\displaystyle \frac{t}{2\sqrt{2}}}))\right|\le {\displaystyle \frac{1}{16}}\left|x-y\right|$$

and

$${\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\Gamma }_{p,q}(\alpha )}}{\int }_0^{{\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}}{({\displaystyle \frac{\mathcal{T}}{p^{\alpha }}}-qr)}^{(\alpha -1)}\mathcal{L}({\displaystyle \frac{r}{p^{\alpha -1}}})d_{p,q}r\approx 0.02553<1.$$

Furthermore, condition $(H_2)$ and (14) are both satisfied. Therefore, by Theorem 2, Equation (18) has at least one non-trivial solution on $t\in [0,2\sqrt{2}]$.

Example 3.

Let $f\in \mathcal{C}([0,2\sqrt{2}]\times \mathbb{R}\to \mathbb{R})$. We study the following $(p,q)$-difference equation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{3/2}x(t)=\frac{1}{10}\left|cos(\frac{t}{2\sqrt{2}})sin(x(\frac{t}{2\sqrt{2}}))\right|,t\in [0,2\sqrt{2}],}\hfill \\ {\displaystyle \frac{3}{4}x(0)+\frac{3}{5}{\mathcal{D}}_{p,q}x(0)=\frac{1}{2},}\hfill \\ {\displaystyle \frac{1}{6}x(2\sqrt{2})+\frac{3}{4}{\mathcal{D}}_{p,q}x(2\sqrt{2})=\frac{1}{3}.}\hfill \end{array}\right.\end{array}\end{array}$$

(19)

Choose $\alpha ={\displaystyle \frac{3}{2}}$, $p={\displaystyle \frac{1}{2}}$, $q={\displaystyle \frac{1}{4}}$, $\mathcal{T}=1$, $a_1={\displaystyle \frac{3}{4}}$, $b_1={\displaystyle \frac{3}{5}}$, $c_1={\displaystyle \frac{1}{2}}$, $a_2={\displaystyle \frac{1}{6}}$, $b_2={\displaystyle \frac{3}{4}}$, $c_2={\displaystyle \frac{1}{3}}$. After simple calculations, we have ${\Lambda }_1\approx 0.3256$, ${\Lambda }_2\approx 0.5039$, $\Lambda \approx 0.5758$, ${\Omega }_1\approx 0.1884$, ${\Omega }_2\approx 0.8477$, $\Phi \approx 6.6032$.

Note that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|f(t,x)\right|& \le {\displaystyle \frac{1}{10}}\left|cos({\displaystyle \frac{t}{2\sqrt{2}}})sin(\parallel x\parallel )\right|\hfill \\ & \le {\displaystyle \frac{1}{10}}\parallel x\parallel .\hfill \end{array}\end{array}$$

Apparently, from (16), we have $\zeta ={\displaystyle \frac{1}{10}}$, $\mathcal{F}(\mu )=\mu$. Based on the above data, we calculate that $\mu >1.5599$. Therefore, $(H_3)$ and $(H_4)$ in Theorem 3 are satisfied. Then, Equation (19) has a non-trivial unique solution on $t\in [0,2\sqrt{2}]$.

5. Applications

In this section, we study a class of models in thermodynamics equations. These models consist of the following equations with the structure of the Jacobi theta function:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}^{\mathcal{C}}{\mathcal{D}}_{p,q}^{1.5}\vartheta (t)=e^{-\pi p^{1.5}t}\vartheta (p^{1.5}t)+\sum _{n=0}^{\infty }{\displaystyle \frac{{(q/p)}^n}{{[n]}_{p,q}!}}{t}^n,t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{1.5}}}]\end{array}\end{array}$$

(20)

and the boundary condition

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\vartheta (0)=1,\hfill \\ {\displaystyle {\mathcal{D}}_{p,q}\vartheta \left(\frac{\mathcal{T}}{p^{1.5}}\right)=\sum _{k\in \mathbb{Z}}{e}^{-\pi k^2{\mathcal{T}}^2}.}\hfill \end{array}\right.\end{array}\end{array}$$

(21)

The fractional derivative on the left side of the equal sign in (20) can describe nonlocal processes with a memory effect, while the first term on the right side can be used to characterize multi-scale decay, where $p^{1.5}$ is the time compression factor. The series term of the second term on the right side can represent the superposition of discrete heat sources or signals, and ${(q/p)}^n$ is used to characterize the energy distribution between scales. The boundary condition (21) can be used to represent the temperature distribution at thermal equilibrium, in which the classical Jacobi theta function represents the periodic thermal radiation [47,48].

Equations (20) and (21) are a special case of Problem (1). We use the Banach fixed-point theorem to prove the existence of solutions to Problem (20) and (21). Here, we omit details of some definitions and applications of the relevant Jacobi theta function, for which readers can refer to [49,50] and references therein. The classical Jacobi theta function satisfies the heat equation ${\displaystyle \frac{\partial \vartheta }{\partial t}}={\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{{\partial }^2\vartheta }{\partial z^2}}$. Our $(p,q)$-deformation generalizes this by replacing the integer-order derivative with a fractional $(p,q)$-difference operator.

In the following, we consider the continuity of the function $f(t,x)=e^{-\pi p^{1.5}t}x+{\sum }_{n=0}^{\infty }{\displaystyle \frac{{(q/p)}^n}{{[n]}_{p,q}!}}{t}^n$. It is easy to know that the exponential function $e^{-\pi p^{1.5}t}$ is continuous for $t\in [0,{\displaystyle \frac{\mathcal{T}}{p^{1.5}}}]$. According to the ratio test and the definition of ${[n]}_{p,q}$, it can be known that the series on the right side is uniformly convergent within any finite interval $[0,{\displaystyle \frac{\mathcal{T}}{p^{1.5}}}]$. Then, $f\in \mathcal{C}([0,{\displaystyle \frac{\mathcal{T}}{p^{1.5}}}]\times \mathbb{R},\mathbb{R})$.

Next, for any $x_1,x_2\in \mathbb{R}$, we have

$$|f(t,x_1)-f(t,x_2)|=\left|e^{-\pi p^{1.5}t}(x_1-x_2)\right|\le e^{-\pi p^{1.5}·0}|x_1-x_2|=1·|x_1-x_2|,$$

and by Theorem 1 and Corollary 1, we can obtain $\mathcal{L}=1$. Taking $\alpha =1.5$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill 1·{\displaystyle \frac{{\mathcal{T}}^{1.5}}{\Gamma (2.5)}}<1\Rightarrow \mathcal{T}<{\left(\Gamma (2.5)\right)}^{2/3}\approx {(1.32934)}^{2/3}\approx 1.21.\end{array}\end{array}$$

According to Banach’s fixed-point theorem, when $\mathcal{T}<1.21$, there exists a unique solution for (20) and (21).

In particular, when $p\to 1^{-}$ and $q\to 1^{-}$, ${[n]}_{p,q}!\to n!$, ${(q/p)}^n\to 1$, $e^{-\pi p^{1.5}t}$, from Definition (5), the $(p,q)$-difference degenerates into the standard difference equation, as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0{}^C{D}_t^{1.5}\vartheta (t)=e^{-\pi t}\vartheta (t)+\sum _{n=0}^{\infty }{\displaystyle \frac{t^n}{n!}}.\end{array}\end{array}$$

Moreover, the series on the right side degenerates into $e^t$; then, the equation can be simplified to

$${}_0{}^C{D}_t^{1.5}\vartheta (t)=e^{-\pi t}\vartheta (t)+e^t.$$

(22)

The degenerated expression (22) corresponds to the fractional heat equation used to simulate anomalous heat diffusion. For specific applications, please refer to [49,50] and references therein.

6. Conclusions

In this paper, we aim to investigate the existence results of non-trivial solutions for Caputo-type fractional $(p,q)$-difference equations under Robin boundary conditions. The existence of solutions is established by using Banach’s fixed-point theorem, Krasnoselskii’s fixed-point theorem and Leray-Schauder’s nonlinear alternative, with the uniqueness of the solutions deduced using Banach’s fixed-point theorem. Finally, we provide three examples to verify our expected results. In addition, the Robin boundary value conditions under study are controlled by $a_i,b_i,c_i(i=1,2)$, which enriches the research topic involving $(p,q)$-difference equations. Moreover, we present the relevant applications of Problem (1) in the field of thermodynamics, which enhances the connection between mathematics and other disciplines. The current research in this paper focuses on a single equation. Future research could attempt to extend this study to a $(p,q)$-difference coupled system in order to investigate the co-existence of multiple solutions.