Upper and lower solutions for an integral boundary problem with two different orders $\left ( p,q\right )$-fractional difference

Abstract

In this paper, a $\left ( p,q\right )$-fractional nonlinear difference equation of different orders is considered and discussed. With the help of $\left ( p,q\right )$-calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.

1 Introduction

In 1910, F.H. Jackson launched an important study in q-calculus and introduced the q-derivative and q-integral, which can be found in [12, 13]. With these results, many researchers joined this field and applied this calculus to several fields, such as orthogonal polynomials, combinatorics, number theory, fundamental hypergeometric functions, mechanics, quantum theory, relativity theory, equations, and derivatives, see [3, 4, 7–11, 14, 16] and the references mentioned therein.

Subsequently, the concept of $\left ( p,q\right )$-calculus underwent generalization and advancement from the q-calculus theory to the two-parameter $\left ( p,q\right )$-integer calculus. This particular mathematical framework has proven to be highly effective across various disciplines. Further information and findings regarding the investigation of $\left ( p,q\right )$-calculus can be explored in [2, 17, 19–27].

Given the importance of the subject of fractional order derivatives and $\left ( p,q\right )$-calculus, many researchers have studied the existence of general or positive solutions, as well as their stability for many problems. So, we present to the reader some references on this matter to enrich the subject of the study [1, 5, 6, 18, 24, 29].

Recently, Qin and Sun [24] conducted a study on positive solutions for a BVP of fractional $\left ( p,q\right )$-difference as presented below

$$ \textstyle\begin{cases} \boldsymbol{D}_{p,q}^{\delta }y\left ( \xi \right ) +h\left ( p^{ \delta }\xi ,y\left ( p^{\delta }\xi \right ) \right ) =0,\quad \xi \in \left ( 0,1\right ), \\ y\left ( 0\right ) =y\left ( 1\right ) =0,\end{cases} $$

(1.1)

where $0< q< p\leq 1$, $1<\delta \leq 2$, $\boldsymbol{D}_{p,q}^{\delta }$ is a Riemann–Liouville-type fractional $\left ( p,q\right )$-difference operator, and $h:\left [ 0,1\right ] \times \mathbb{R} \rightarrow \mathbb{R}$ is a nonnegative continuous function; $\boldsymbol{D}_{p,q}^{\delta }$ denotes the $\left ( p,q\right )$-fractional difference operator of order δ.

Xu and Sun [29] investigated the positivity of a class of integral boundary value problems of fractional differential equations with two nonlinear terms of the form

$$ \textstyle\begin{cases} \boldsymbol{D}^{\delta }y\left ( \xi \right ) +h\left ( \xi ,y\left ( \xi \right ) \right ) =\boldsymbol{D}^{\delta -1}g\left ( \xi ,y \left ( \xi \right ) \right ) ,\quad \xi \in \left ( 0,1\right ) , \\ y\left ( 0\right ) =0,\ \ y\left ( 1\right ) =\int _{0}^{1}g\left ( s,y \left ( s\right ) \right ) \mathrm{d}s,\end{cases} $$

(1.2)

where $\boldsymbol{D}^{\delta }$ is the standard Riemann–Liouville derivative, $1<\delta \leq 2$.

In 2018, Xu and Han [28] used the method of upper and lower solutions and studied the problem

$$ \textstyle\begin{cases} \boldsymbol{D}^{\delta }y\left ( \xi \right ) +h\left ( \xi ,y\left ( \xi \right ) \right ) =\boldsymbol{D}^{\tau }g\left ( \xi ,y\left ( \xi \right ) \right ) ,\quad \xi \in \left ( 0,1\right ) , \\ y\left ( 0\right ) =0,\ \ y\left ( 1\right ) = \frac{1}{\Gamma \left ( \delta -\tau \right ) }\int _{0}^{1}\left ( 1-s \right ) ^{\left ( \delta -\tau -1\right ) }g\left ( s,y\left ( s \right ) \right ) \mathrm{d}s,\end{cases} $$

(1.3)

where $\boldsymbol{D}^{\delta }$ and $\boldsymbol{D}^{\tau }$ are the standard Riemann–Liouville derivatives, $1<\delta \leq 2$, $0<\tau <\delta$.

The aim of this paper is to investigate an integral boundary value problem of $\left ( p,q\right )$-fractional difference equations which encompasses two nonlinear terms defined as follows:

$$ \textstyle\begin{cases} \boldsymbol{D}_{p,q}^{\delta }y\left ( \xi \right ) +h\left ( p^{ \delta }\xi ,y\left ( p^{\delta }\xi \right ) \right ) = \boldsymbol{D}_{p,q}^{\tau }g\left ( p^{\delta -\tau }\xi ,y\left ( p^{ \delta -\tau }\xi \right ) \right ) ,\quad \xi \in \left ( 0,1\right ) , \\ y\left ( 0\right ) =0,\ \ y\left ( 1\right ) = \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{1}\left ( 1-qv \right ) ^{\delta -\tau -1}g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu ,\end{cases} $$

(1.4)

where $0< q< p\leq 1$, $\delta \in \left ( 1,2\right ]$, $0<\tau <\delta$, the functions $h,g:\left [ 0,1\right ] \times \left [ 0,\infty \right ) \rightarrow \left [ 0,\infty \right )$ are continuous with $g\left ( \xi ,y\right )$ being nondecreasing in y, and where $p^{\binom{\delta -\tau }{2}}:= \frac{\left ( \delta -\tau \right ) \left ( \delta -\tau -1\right ) }{2}$.

As far as we are aware, no publication has examined the possibility of finding positive solutions for the $\left ( p,q\right )$-fractional difference equations with integral boundary conditions and two nonlinear terms with different fractional order derivatives, by constructing upper and lower control functions and then giving some existence and uniqueness results using Schauder fixed point theorem and Banach contracting mapping principle. On the other hand, this problem is more general than the others in the literature.

This paper is produced as follows. In Sect. 2, we give some necessary concepts, tools, and results used in the analysis. Section 3 is devoted to studying the existence of positive solutions. An example is given in Sect. 4. Some conclusions and generalizations are drawn in Sect. 5.

2 Essential materials

This part presents some essential material, which is required for our study. We begin with some fundamental definitions and results of the q-calculus and $\left ( p,q\right )$-calculus, which can be found in [9, 25–27]. Let $0< q< p\leq 1$ be constants,

$$\begin{aligned}& \left [ m\right ] _{p,q}:= \textstyle\begin{cases} \frac{p^{m}-q^{m}}{p-q}=p^{m-1}\left [ m\right ] _{\frac{q}{p}}, & m \in \mathbb{N} ^{+}, \\ 1, & m=0,\end{cases}\displaystyle \\& \left [ m\right ] _{p,q}!:= \textstyle\begin{cases} \left [ m\right ] _{p,q}\left [ m-1\right ] _{p,q}\cdots \left [ 1 \right ] _{p,q}=\prod \limits _{i=1}^{m}\frac{p^{i}-q^{i}}{p-q}, & m \in \mathbb{N} ^{+}, \\ 1, & m=0.\end{cases}\displaystyle \end{aligned}$$

The q-analogue of the power function $\left ( a-b\right ) _{q}^{\left ( n\right ) }$ is given by

$$ \left ( a-b\right ) _{q}^{\left ( n\right ) }:= \textstyle\begin{cases} \prod \limits _{i=0}^{n-1}\left ( a-bq^{i}\right ) , & n\in \mathbb{N} ^{+},\ a,b\in \mathbb{R}\mathbbm{,} \\ 1, & n=0.\end{cases} $$

The $\left ( p,q\right )$-analogue of the power function $\left ( a-b\right ) _{p,q}^{\left ( n\right ) }$ is given by

$$ \left ( a-b\right ) _{p,q}^{\left ( n\right ) }:= \textstyle\begin{cases} \prod \limits _{i=0}^{n-1}\left ( ap^{i}-bq^{i}\right ) , & n\in \mathbb{N} ^{+},\ a,b\in \mathbb{R}\mathbbm{,} \\ 1, & n=0,\end{cases} $$

and for $\delta \in \mathbb{R}$, the general form of the above is given by

$$ \left ( a-b\right ) _{p,q}^{\left ( \delta \right ) }:=p^{ \binom{\delta }{2} }\left ( a-b\right ) _{\frac{q}{p}}^{\left ( \delta \right ) }=a^{\delta }p^{ \binom{\delta }{2}}\prod \limits _{i=0}^{ \infty } \frac{a-b\left ( \frac{q}{p} \right ) ^{i}}{a-b\left ( \frac{q}{p}\right ) ^{\delta +i}},\quad \ 0< b< a, $$

where $p^{\binom{\delta }{2}}:=\frac{\delta \left ( \delta -1\right ) }{2}$.

Definition 1

([26])

Let $0< q< p\leq 1$. The $\left ( p,q\right )$-derivative of a function h is defined as

$$ \boldsymbol{D}_{p,q}h\left ( \xi \right ) := \frac{h\left ( p\xi \right ) -h\left ( q\xi \right ) }{\left ( p-q\right ) \xi },\quad \xi \neq 0, $$

and $\left ( \boldsymbol{D}_{p,q}h\right ) \left ( 0\right ) =\lim _{\xi \rightarrow 0}\left ( \boldsymbol{D}_{p,q}h\right ) \left ( \xi \right )$, whenever h is differentiable at 0. Moreover, the high order $\left ( p,q\right )$-derivative $\boldsymbol{D}_{p,q}^{n}h\left ( \xi \right )$ is defined by

$$ \boldsymbol{D}_{p,q}^{n}h\left ( \xi \right ) = \textstyle\begin{cases} \boldsymbol{D}_{p,q}\boldsymbol{D}_{p,q}^{n-1}h\left ( \xi \right ) , & n\in \mathbb{N} ^{+}, \\ h\left ( \xi \right ) , & n=0.\end{cases} $$

Definition 2

([26])

Let $0< q< p\leq 1$ and h be an arbitrary function of a real variable ξ. The $\left ( p,q\right )$-integral of the function h is defined as

$$ \boldsymbol{I}_{p,q}h\left ( \xi \right ) :=\int _{0}^{\xi }h\left ( \nu \right ) \mathrm{d}_{p,q}\nu =\left ( p-q\right ) \xi \sum _{i=0}^{ \infty }\frac{q^{i}}{ p^{i+1}}h\left ( \frac{q^{i}}{p^{i+1}}\xi \right ) , $$

as long as the series on the right-hand side converges. In this case, h is called $\left ( p,q\right )$-integrable on $\left [ 0,\xi \right ]$.

Definition 3

([27])

Let $\delta >0$, $0< q< p\leq 1$, and $h:\left [ 0,1\right ] \rightarrow \mathbb{R}$ be an arbitrary function. The fractional $\left ( p,q\right )$-integral of order δ is defined by

$$ \boldsymbol{I}_{p,q}^{\delta }h\left ( \xi \right ) = \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) }\int _{0}^{ \xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }h \left ( \frac{\nu }{p^{\delta -1}}\right ) \mathrm{d}_{p,q}\nu , $$

with $p^{\binom{\delta }{2}}:=\frac{\delta \left ( \delta -1\right ) }{2}$ and $\boldsymbol{I}_{p,q}^{0}h\left ( \xi \right ) =h\left ( \xi \right )$.

Definition 4

([27])

Let $\delta >0$, $0< q< p\leq 1$, and h be an arbitrary function on $\left [ 0,1\right ]$. The fractional $\left ( p,q\right )$-difference operator of Riemann–Liouville type of order δ is defined by

$$ \boldsymbol{D}_{p,q}^{\delta }h\left ( \xi \right ) =\boldsymbol{D}_{p,q}^{m} \boldsymbol{I}_{p,q}^{m-\delta }h\left ( \xi \right ) $$

and $\boldsymbol{D}_{p,q}^{0}h\left ( \xi \right ) =h\left ( \xi \right )$, where m is the smallest integer greater than or equal to δ. In addition,

$$ \boldsymbol{D}_{p,q}^{\delta }\boldsymbol{I}_{p,q}^{\delta }h\left ( \xi \right ) =h\left ( \xi \right ) . $$

Lemma 1

([27])

For $\delta \in \left ( m-1,m\right ]$, $m\in \mathbb{N}$, $0< q< p\leq 1$, and $h:\left [ 0,1\right ] \rightarrow \mathbb{R}$. We get

$$ \boldsymbol{I}_{p,q}^{\delta }\boldsymbol{D}_{p,q}^{\delta }h\left ( \xi \right ) =h\left ( \xi \right ) +c_{1}\xi ^{\delta -1}+c_{2}\xi ^{ \delta -2}+\cdots +c_{N}\xi ^{\delta -N}, $$

for some $c_{j}\in \mathbb{R}$, $j=1,2,\dots ,m$.

3 Main results

In this part, we shall demonstrate the positivity of solutions for boundary value problems (1.4) by employing Schauder fixed point theorem, and then establish sufficient conditions for uniqueness using Banach fixed point theorem. We will utilize the method of ULS in the analysis.

Lemma 2

Let $0< q< p\leq 1$, $1<\delta \leq 2$, and $0<\tau <\delta$. Then $y\left ( \xi \right )$ is a solution of the IBVP (1.4) if and only if $y\left ( \xi \right )$ satisfies the integral equation

$$ \begin{aligned} y\left ( \xi \right ) &=\int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu , \end{aligned} $$

where

$$ \boldsymbol{G}\left ( \xi ,q\nu \right ) = \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \textstyle\begin{cases} \xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }-\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }, & 0\leq q\nu \leq \xi \leq 1, \\ \xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }, & 0\leq \xi \leq q\nu \leq 1,\end{cases} $$

(3.1)

Proof

The proof is divided into two cases.

Case 1. $\tau \leq 1$. From Definition 3 and Lemma 1, applying the operator $\boldsymbol{I}_{p,q}^{\delta }$ on both sides of (1.4), one has

$$ \begin{aligned} &y\left ( \xi \right ) +c_{1}\xi ^{\delta -1}+c_{2}\xi ^{ \delta -2}+ \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) }\int _{0}^{\xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &=\boldsymbol{I}_{p,q}^{\delta -\tau }\left ( \boldsymbol{I}_{p,q}^{ \tau }\boldsymbol{D}_{p,q}^{\tau }g\left ( p\xi ,y\left ( p\xi \right ) \right ) \right ) \\ &=\boldsymbol{I}_{p,q}^{\delta -\tau }\left ( g\left ( p\xi ,y\left ( p \xi \right ) \right ) +c_{3}\xi ^{\tau -1}\right ) \\ &= \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta - \tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu +c_{3} \frac{p^{\binom{\tau }{2}}\Gamma _{p,q}\left ( \tau \right ) }{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\xi ^{ \delta -1}, \end{aligned} $$

for some constants $c_{1},c_{2},c_{3}\in \mathbb{R}$. Due to the boundary condition $y\left ( 0\right ) =0$, and the last formula, we get $c_{2}=0$ and, since

$$ y\left ( 1\right ) = \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{1}\left ( 1-qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu , $$

obtain

$$ c_{1}=- \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{0}^{1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu +c_{3} \frac{p^{\binom{\tau }{2}}\Gamma _{p,q}\left ( \tau \right ) }{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }. $$

Case 2. $\tau >1$. As in Case 1, we can write

$$ \begin{aligned} &y\left ( \xi \right ) +c_{1}\xi ^{\delta -1}+c_{2}\xi ^{ \delta -2}+ \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) }\int _{0}^{\xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &=\boldsymbol{I}_{p,q}^{\delta -\tau }\left ( g\left ( p\xi ,y\left ( p \xi \right ) \right ) +c_{3}\xi ^{\tau -1}+c_{4}\xi ^{\tau -2}\right ) \\ &= \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta - \tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad +c_{3} \frac{p^{\binom{\tau }{2}}\Gamma _{p,q}\left ( \tau \right ) }{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\xi ^{ \delta -1}+c_{4}\frac{p^{\binom{\tau -1}{2}}\Gamma _{p,q}\left ( \tau -1\right ) }{p^{\binom{\delta -1}{2}}\Gamma _{p,q}\left ( \delta -1\right ) }\xi ^{\delta -2}, \end{aligned} $$

for some constants $c_{1},c_{2},c_{3},c_{4}\in \mathbb{R}$. Due to the boundary conditions, we get $c_{2}= c_{4} \frac{p^{\binom{\tau -1}{2}}\Gamma _{p,q}\left ( \tau -1\right ) }{p^{\binom{\delta -1}{2}}\Gamma _{p,q}\left ( \delta -1\right ) }$ and

$$ c_{1}=- \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{0}^{1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu +c_{3} \frac{p^{\binom{\tau }{2}}\Gamma _{p,q}\left ( \tau \right ) }{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }. $$

Therefore,

$$ \begin{aligned} y\left ( \xi \right ) &= \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{0}^{1}\xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad - \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{0}^{\xi }\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1 \right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &= \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{0}^{\xi }\left [ \xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }-\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }\right ] h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{\ d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) } \int _{\xi }^{1}\xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &=\int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) h\left ( p \nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu . \end{aligned} $$

The proof is finished, and this process is reversible. □

Lemma 3

([24])

The function G defined by (3.1) satisfies the following properties:

• (i) $\boldsymbol{G}\left ( \xi ,q\nu \right ) \leq \xi ^{\delta -1}\left ( 1-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }\leq \left ( 1-q \nu \right ) _{p,q}^{\left ( \delta -1\right ) }$,

• (ii) $0\leq \boldsymbol{G}\left ( \xi ,q\nu \right ) \leq 1$,

for all $0\leq \xi ,\nu \leq 1$.

Let the Banach space $\boldsymbol{E}=C\left ( \left [ 0,1\right ] \right )$ be endowed with the norm $\left \Vert y\right \Vert =\max _{\xi \in \left [ 0,1 \right ] } \left \vert y\left ( \xi \right ) \right \vert$. Denote

$$ \boldsymbol{P}=\left \{ y\in \boldsymbol{E}:y\left ( \xi \right ) \geq 0,\xi \in \left [ 0,1\right ] \right \} . $$

Let a and b be positive real numbers, where b is greater than a. For any y belonging to the interval $\left [ a,b\right ]$, we define the lower control function

$$ V\left ( \xi ,y\right ) =\inf \left \{ h\left ( \xi ,\lambda \right ) :y \leq \lambda \leq b\right \} $$

and the upper control function

$$ W\left ( \xi ,y\right ) =\sup \left \{ h\left ( \xi ,\lambda \right ) :a \leq \lambda \leq y\right \} . $$

Obviously, $V\left ( \xi ,y\right )$ and $W\left ( \xi ,y\right )$ are monotonous and nondecreasing in y, and

$$ V\left ( \xi ,y\right ) \leq h\left ( \xi ,y\right ) \leq W\left ( \xi ,y\right ) . $$

Now, let the functions $\alpha \left ( \xi \right )$ and $\beta \left ( \xi \right )$ be the lower and upper solutions of (1.4), respectively. We need the following hypotheses:

$$ \begin{aligned} \alpha \left ( \xi \right ) &\leq \int _{0}^{1} \boldsymbol{G}\left ( \xi ,q\nu \right ) V\left ( p\nu ,\alpha \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,\alpha \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \end{aligned} $$

(3.2)

and

$$ \begin{aligned} \beta \left ( \xi \right ) &\geq \int _{0}^{1} \boldsymbol{G}\left ( \xi ,q\nu \right ) W\left ( p\nu ,\beta \left ( p \nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,\beta \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu , \end{aligned} $$

(3.3)

for all $\xi \in \left [ 0,1\right ]$.

Theorem 1

Let (3.2) and (3.3) hold. Then (1.4) has at least one positive solution $y\in \boldsymbol{E}$ and $\alpha \left ( \xi \right ) \leq y\left ( \xi \right ) \leq \beta \left ( \xi \right )$, $\xi \in \left [ 0,1\right ]$.

Proof

Let

$$ \boldsymbol{K}=\{y\in \boldsymbol{P}:\alpha \left ( \xi \right ) \leq y\left ( \xi \right ) \leq \beta \left ( \xi \right ) ,\xi \in \left [ 0,1\right ] \}. $$

It is evident that $\left \Vert y\right \Vert \leq b$. Consequently, $\boldsymbol{K}\subset \boldsymbol{E}$ is convex, bounded, and closed. If $y\in \boldsymbol{K}$, there exist positive constants $M_{h}$ and $M_{g}$ such that

$$\begin{aligned} \max _{\xi \in \left [ 0,1\right ] }h\left ( p\nu ,y\left ( p\nu \right ) \right ) & < M_{h}, \\ \max _{\xi \in \left [ 0,1\right ] }g\left ( p\nu ,y\left ( p\nu \right ) \right ) &< M_{g}. \end{aligned}$$

We define the operator ϕ as

$$ \begin{aligned} \phi y\left ( \xi \right ) &=\int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu . \end{aligned} $$

(3.4)

The above operator ϕ is continuous on K due to the continuity of h and g.

If $y\in \boldsymbol{K}$, we can obtain

$$ \begin{aligned} \phi y\left ( \xi \right ) &=\int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) \left \vert h\left ( p\nu ,y\left ( p\nu \right ) \right ) \right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\left \vert g\left ( p\nu ,y\left ( p\nu \right ) \right ) \right \vert \mathrm{d}_{p,q}\nu \\ &\leq M_{h}\int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \mathrm{d} _{p,q}\nu + \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\mathrm{d}_{p,q}\nu , \end{aligned} $$

where, due to Lemma 3, $0\leq \boldsymbol{G}\left ( \xi ,q\nu \right ) \leq 1$, and then we get

$$ \phi y\left ( \xi \right ) \leq M_{h}+ \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) }. $$

Thus $\phi \left ( \boldsymbol{K}\right )$ is uniformly bounded.

Next, for each $y\in \boldsymbol{K}$, $\xi _{1},\xi _{2}\in \left [ 0,1\right ]$, $\xi _{1}<\xi _{2}$, we get

$$ \begin{aligned} &\left \vert \phi y\left ( \xi _{2}\right ) -\phi y \left ( \xi _{1}\right ) \right \vert \\ &=\Bigg\vert \int _{0}^{1}\boldsymbol{G}\left ( \xi _{2},q\nu \right ) h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\qquad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{ \xi _{2}}\left ( \xi _{2}-qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\qquad - \int _{0}^{1}\boldsymbol{G}\left ( \xi _{1},q\nu \right ) h \left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\qquad - \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{ \xi _{1}}\left ( \xi _{1}-qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \Bigg\vert \\ &\leq M_{h}\int _{0}^{1}\left \vert \boldsymbol{G}\left ( \xi _{2},q \nu \right ) -\boldsymbol{G}\left ( \xi _{1},q\nu \right ) \right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \Bigg\vert \int _{0}^{\xi _{2}}\left ( \xi _{2}-qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\qquad \qquad \qquad \qquad \qquad \quad -\int _{0}^{\xi _{1}}\left ( \xi _{1}-qv\right ) ^{\left ( \delta -\tau -1\right ) }g\left ( p\nu ,y \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \Bigg\vert \\ &\leq \frac{M_{h}}{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\int _{0}^{\xi _{1}}\left \vert \left ( 1-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }\left ( \xi _{2}^{\delta -1}-\xi _{1}^{ \delta -1}\right ) -\left ( \xi _{2}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }+\left ( \xi _{1}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }\right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{M_{h}}{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\int _{\xi _{1}}^{\xi _{2}}\left \vert \left ( 1-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }\left ( \xi _{2}^{\delta -1}-\xi _{1}^{ \delta -1}\right ) -\left ( \xi _{2}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }\right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{M_{h}}{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\int _{\xi _{2}}^{1}\left \vert \left ( 1-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }\left ( \xi _{2}^{\delta -1}-\xi _{1}^{ \delta -1}\right ) \right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi _{2}}\left \vert \left ( \xi _{2}-qv\right ) ^{\left ( \delta -\tau -1\right ) }-\left ( \xi _{1}-qv\right ) ^{\left ( \delta -\tau -1\right ) }\right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{t_{2}}^{\xi _{1}}\left ( \xi _{1}-qv\right ) ^{\left ( \delta - \tau -1\right ) }\mathrm{d}_{p,q}\nu . \end{aligned} $$

Thus,

$$\begin{aligned} &\left \vert \phi y\left ( \xi _{2}\right ) -\phi y \left ( \xi _{1}\right ) \right \vert \\ &\leq \frac{M_{h}}{p^{\left ( _{2}^{\delta }\right ) }\Gamma _{p,q}\left ( \delta \right ) }\bigg\{ \int _{0}^{\xi _{1}}\left ( \xi _{2}^{ \delta -1}-\xi _{1}^{\delta -1}\right ) \mathrm{d}_{p,q}\nu +\int _{0}^{ \xi _{1}}\left [ \left ( \xi _{2}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }-\left ( \xi _{1}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }\right ] \mathrm{d}_{p,q}\nu \\ &\qquad \qquad \qquad \quad + \int _{\xi _{1}}^{\xi _{2}}\left ( \xi _{2}^{ \delta -1}-\xi _{1}^{\delta -1}+1\right ) \mathrm{d}_{p,q}\nu +\int _{ \xi _{2}}^{1}\left ( \xi _{2}^{\delta -1}-\xi _{1}^{\delta -1}\right ) \mathrm{d}_{p,q}\nu \bigg\} \\ &\quad + \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \bigg\{ \int _{0}^{\xi _{2}}\left \vert \left ( \xi _{2}-qv\right ) ^{ \left ( \delta -\tau -1\right ) }-\left ( \xi _{1}-qv\right ) ^{ \left ( \delta -\tau -1\right ) }\right \vert \mathrm{d}_{p,q}\nu \\ &\qquad \qquad \qquad \qquad \qquad \quad +\int _{t_{2}}^{\xi _{1}} \left ( \xi _{1}-qv\right ) ^{\delta -\tau -1}\mathrm{d}_{p,q}\nu \bigg\} \\ &\leq \frac{M_{h}}{p^{\binom{\delta }{2}}\Gamma _{p,q}\left ( \delta \right ) }\bigg\{ \int _{0}^{1}\left ( \left [ \left ( \xi _{2}-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }-\left ( \xi _{1}-q\nu \right ) _{p,q}^{ \left ( \delta -1\right ) }\right ] +2\left ( \xi _{2}-\xi _{1} \right ) ^{\delta -1}\right ) \mathrm{d}_{p,q}\nu \\ &\qquad \qquad \qquad \ \ + \int _{\xi _{1}}^{\xi _{2}}\left ( \xi _{2}- \xi _{1}\right ) ^{\delta -1}\mathrm{d}_{p,q}\nu +\xi _{2}-\xi _{1} \bigg\} \\ &\quad + \frac{M_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \bigg\{ \int _{0}^{\xi _{2}}\left \vert \left ( \xi _{2}-qv\right ) ^{ \left ( \delta -\tau -1\right ) }-\left ( \xi _{1}-qv\right ) ^{ \left ( \delta -\tau -1\right ) }\right \vert \mathrm{d}_{p,q}\nu \\ &\qquad \qquad \qquad \qquad \qquad \quad +\int _{t_{2}}^{\xi _{1}} \left ( \xi _{1}-qv\right ) ^{\delta -\tau -1}\mathrm{d}_{p,q}\nu \bigg\} . \end{aligned}$$

(3.5)

Since the function $\left ( \xi -q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }$ is continuous with respect to ξ and ν on $\left [ 0,1\right ] \times \left [ 0,1\right ]$, it can be inferred that the function is uniformly continuous on $\left [ 0,1\right ] \times \left [ 0,1\right ]$. Consequently, for any $\nu \in \left [ 0,1\right ]$, we can deduce the following:

$$ \left ( \xi _{2}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) }- \left ( \xi _{1}-q\nu \right ) _{p,q}^{\left ( \delta -1\right ) } \rightrightarrows 0\quad \text{as}\ \ \xi _{1}\rightarrow \xi _{2} $$

and

$$ \left ( \xi _{2}-qv\right ) ^{\left ( \delta -\tau -1\right ) }- \left ( \xi _{1}-qv\right ) ^{\left ( \delta -\tau -1\right ) } \rightrightarrows 0\quad \text{as}\ \ \xi _{1}\rightarrow \xi _{2}. $$

So we can say that, as $\xi _{1}\rightarrow \xi _{2}$, the right-hand side of the inequality (3.5) converges to zero. As a result, $\phi \left ( \boldsymbol{K}\right )$ exhibits equicontinuity on the interval $\left [ 0,1\right ]$. By utilizing Arzela–Ascoli theorem, it can be concluded that $\phi :\boldsymbol{K}\rightarrow \boldsymbol{E}$ is compact.

We will now demonstrate that $\phi \left ( \boldsymbol{K}\right ) \subset \boldsymbol{K}$. Consider an element $y\in \boldsymbol{K}$. It can be inferred from (3.2) that

$$\begin{aligned} \phi y\left ( \xi \right ) &=\int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) h\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &\leq \int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) W\left ( p \nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu \\ &\leq \int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) W\left ( p \nu ,\beta \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,\beta \left ( p\nu \right ) \right ) \mathrm{d} _{p,q}\nu \\ &\leq \beta \left ( \xi \right ) , \end{aligned}$$

and, in the same way, we also have

$$ \begin{aligned} \phi y\left ( \xi \right ) &\geq \int _{0}^{1} \boldsymbol{G}\left ( \xi ,q\nu \right ) V\left ( p\nu ,\alpha \left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,\alpha \left ( p\nu \right ) \right ) \mathrm{d} _{p,q}\nu \\ &\geq \alpha \left ( \xi \right ) . \end{aligned} $$

Using Schauder fixed point theorem, it can be concluded that ϕ possesses at least one fixed point y within K. Consequently, equation (1.4) will have at least one positive solution y within E, satisfying the conditions $\alpha \left ( \xi \right ) \leq y\left ( \xi \right ) \leq \beta \left ( \xi \right )$ for all $\xi \in \left [ 0,1\right ]$. □

Corollary 1

Suppose there are nonnegative bounded continuous functions $\kappa _{1}$, $\kappa _{2}$, $\kappa _{3}$, and $\kappa _{4}$ such that

$$\begin{aligned} \kappa _{1}\left ( \xi \right ) &\leq h\left ( \xi ,y\right ) \leq \kappa _{2}\left ( \xi \right ) ,\ \ \left ( \xi ,y\right ) \times \left [ 0,1\right ] \times \left [ 0,\infty \right ) , \end{aligned}$$

(3.6)

$$\begin{aligned} \kappa _{3}\left ( \xi \right ) &\leq g\left ( \xi ,y\right ) \leq \kappa _{4}\left ( \xi \right ) ,\ \ \left ( \xi ,y\right ) \times \left [ 0,1\right ] \times \left [ 0,\infty \right ) , \end{aligned}$$

(3.7)

and at least one of $\kappa _{1}\left ( \xi \right )$ and $\kappa _{3}\left ( \xi \right )$ is not identically equal to 0. Then, equation (1.4) will have at least one positive solution $y\in \boldsymbol{E}$ such that

$$ \begin{aligned} &\int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \kappa _{1}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\kappa _{3}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu \\ &\leq y\left ( \xi \right ) \\ &\leq \int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \kappa _{2} \left ( \nu q\right ) \mathrm{d}_{p,q}\nu + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\kappa _{4}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu , \end{aligned} $$

(3.8)

for $\delta -\tau \geq 1$, and

$$ \begin{aligned} &\int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \kappa _{1}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\kappa _{4}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu \\ &\leq y\left ( \xi \right ) \\ &\leq \int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \kappa _{2} \left ( \nu q\right ) \mathrm{d}_{p,q}\nu + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\kappa _{3}\left ( \nu q\right ) \mathrm{d}_{p,q}\nu , \end{aligned} $$

(3.9)

for $\delta -\tau \leq 1$.

Proof

Consider the problem

$$ \textstyle\begin{cases} \boldsymbol{D}_{p,q}^{\delta }y\left ( \xi \right ) +\kappa _{2} \left ( p^{\delta }\xi \right ) =\boldsymbol{D}_{p,q}^{\tau }\kappa _{4} \left ( p^{\delta -1}\xi \right ) ,\quad \xi \in \left ( 0,1\right ) , \\ y\left ( 0\right ) =0,\ \ y\left ( 1\right ) = \frac{1}{p^{\binom{\delta -\tau }{2 }}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{1}\left ( 1-qv \right ) ^{\left ( \delta -\tau -1\right ) }\kappa _{4}\left ( p\nu \right ) \mathrm{d} _{p,q}\nu ,\end{cases} $$

(3.10)

which is equivalent to

$$ y\left ( \xi \right ) =\int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \kappa _{2}\left ( p\nu \right ) \mathrm{d}_{p,q}\nu + \frac{1}{p^{\binom{ \delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) }\int _{0}^{ \xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1\right ) } \kappa _{4}\left ( p\nu \right ) \mathrm{d}_{p,q}\nu , $$

(3.11)

where G is given by (3.1). According to the control function definitions, we have

$$ \kappa _{1}\left ( \xi \right ) \leq V\left ( \xi ,y\right ) \leq W \left ( \xi ,y\right ) \leq \kappa _{2}\left ( \xi \right ) ,\quad \left ( \xi ,y\right ) \times \left [ 0,1\right ] \times \left [ a,b \right ] , $$

where a, b are minimal and maximal values of $y\left ( \xi \right )$ on $\left [ 0,1 \right ]$. Hence, we can acquire

$$ \begin{aligned} y\left ( \xi \right ) &\geq \int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) W\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }g\left ( p\nu ,y\left ( p\nu \right ) \right ) \mathrm{d}_{p,q} \nu . \end{aligned} $$

Since $p^{\binom{\delta -\tau }{2}}$ is positive for $\delta -\tau \geq 1$, then (3.11) is an upper solution of (1.4). Similarly, we can demonstrate using the same method that

$$ \begin{aligned} z\left ( \xi \right ) &=\int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) \kappa _{1}\left ( p\nu \right ) \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{p^{\binom{\delta -\tau }{2}}\Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\kappa _{3}\left ( p\nu \right ) \mathrm{d}_{p,q}\nu \end{aligned} $$

is a lower solution of (1.4). Thus the problem (1.4) has at least one positive solution $y\in \boldsymbol{E}$ that satisfies (3.8), according to Theorem 1.

We use the same steps for the case $\delta -\tau \leq 1$ to obtain (3.9), and take into account that $p^{\binom{\delta -\tau }{2}}$ changes the inequality (3.7). □

Theorem 2

Assume (3.2) and (3.3) hold and that there exist constants $L_{h},L_{g}>0$ such that for each $\xi \in \left [ 0,1\right ]$, $x,y\in \boldsymbol{K}$, the inequalities

$$\begin{aligned} \left \vert h\left ( \xi ,x\right ) -h\left ( \xi ,y\right ) \right \vert &\leq L_{h}\left \Vert x-y\right \Vert , \\ \left \vert g\left ( \xi ,x\right ) -g\left ( \xi ,y\right ) \right \vert &\leq L_{g}\left \Vert x-y\right \Vert \end{aligned}$$

are satisfied with $L_{h}+ \frac{L_{g}}{\left \vert p^{\binom{\delta -\tau }{2} }\right \vert \Gamma _{p,q}\left ( \delta -\tau +1\right ) }<1$. Then, the problem (1.4) possesses a unique positive solution on K.

Proof

After establishing the validity of Theorem 1, it becomes evident that $\phi :\boldsymbol{K}\rightarrow \boldsymbol{K}$. Then, for any x and y belonging to K, we get

$$ \begin{aligned} &\left \vert \phi x\left ( \xi \right ) -\phi y\left ( \xi \right ) \right \vert \\ &\leq \int _{0}^{1}\boldsymbol{G}\left ( \xi ,q\nu \right ) \left \vert h\left ( p\nu ,x\left ( p\nu \right ) \right ) -h\left ( p\nu ,y \left ( p\nu \right ) \right ) \right \vert \mathrm{d}_{p,q}\nu \\ &\quad + \frac{1}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \int _{0}^{\xi }\left ( \xi -qv\right ) ^{\left ( \delta -\tau -1 \right ) }\left \vert g\left ( p\nu ,x\left ( p\nu \right ) \right ) -g \left ( p\nu ,y\left ( p\nu \right ) \right ) \right \vert \mathrm{d}_{p,q} \nu \\ &\leq L_{h}\left \Vert x-y\right \Vert \int _{0}^{1}\boldsymbol{G} \left ( \xi ,q\nu \right ) \mathrm{d}_{p,q}\nu + \frac{L_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau \right ) } \left \Vert x-y\right \Vert \int _{0}^{\xi }\left ( \xi -qv\right ) ^{ \left ( \delta -\tau -1\right ) }\mathrm{d}_{p,q}\nu , \end{aligned} $$

where, due to Lemma 3, $0\leq \boldsymbol{G}\left ( \xi ,q\nu \right ) \leq 1$, and then

$$ \left \Vert \phi x-\phi y\right \Vert \leq \left ( L_{h}+ \frac{L_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau +1\right ) }\right ) \left \Vert x-y\right \Vert . $$

Hence, ϕ is a contraction mapping because $L_{h}+ \frac{L_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau +1\right ) }<1$. Thus, there is a fixed point $y\in \boldsymbol{K}$ which serves as the sole positive solution to the problem (1.4) on K. □

4 Example

Now, we give an example illustrating Theorems 1 and 2. Consider the integral boundary value problem (1.4), where $\delta =1.75$, $\tau =0.25$, $p=0.95$, $q=0.5$, $g\left ( \xi ,y\right ) =1+\xi +\frac{y}{7+y}$, and $h\left ( \xi ,y\right ) =\xi ^{2}+ \frac{\xi ^{2}y\left ( \xi \right ) }{5+y\left ( \xi \right ) }$. It is easy to see that the function g is nondecreasing in y, and

$$\begin{aligned} \xi ^{2}\leq h\left ( \xi ,y\right ) \leq \xi ^{2}+1\leq 2, \text{ for } \left ( \xi ,y\right ) \in \left [ 0,1\right ] \times \left [ 0,\infty \right ) , \\ 1+\xi \leq g\left ( \xi ,y\right ) \leq 2+\xi ,\text{ for }\left ( \xi ,y\right ) \in \left [ 0,1\right ] \times \left [ 0,\infty \right ) . \end{aligned}$$

By Corollary 1, the problem (1.4) has at least one solution. On the other hand, since $L_{h}=\frac{1}{5}$ and $L_{g}=\frac{1}{3}$, we have

$$ L_{h}+ \frac{L_{g}}{\left \vert p^{\binom{\delta -\tau }{2}}\right \vert \Gamma _{p,q}\left ( \delta -\tau +1\right ) }= \frac{1}{5}+\frac{\frac{1}{3}}{\frac{ 1.5\times 0.5}{2}\times 2.6273}\simeq 0.53< 1. $$

According to Theorem 2, the problem (1.4) possesses a unique solution that is positive.

5 Conclusion

Fractional calculus, which involves studying derivatives of arbitrary order, is an important field of research due to its extensive theoretical advancements and practical applications over the past few decades. So, we investigated in this paper a class of integral boundary value problems of $(p,q)$-difference equations with two nonlinear terms containing fractional derivatives. The method of ULS, along with Schauder fixed point theorem, was utilized to obtain positive solutions. Additionally, Banach contraction mapping principle was employed to establish uniqueness results. The results obtained in this paper are good and important, as the existence and uniqueness results in [24] can be obtained by the method used in this research when the function $g\equiv 0$.

A compelling avenue for future research would be to explore fractional $(p,q)$-difference equations with variable orders, as opposed to the constant order examined in this study. Additionally, investigating boundary conditions of the Riemann–Stieltjes integral type presents another promising direction. Incorporating impulsive effects into the analysis would further broaden the scope of potential applications, see [15] and some related works that can be applied.