On the Generalized (p,q)-ϕ-Calculus with Respect to Another Function

Abstract

In the present paper, we generalized some of the operators defined in $(p,q)$-calculus with respect to another function. More precisely, the generalized $(p,q)$-$\varphi$-derivatives and $(p,q)$-$\varphi$-integrals were introduced with respect to the strictly increasing function $\varphi$ with the help of different orders of the q-shifting, p-shifting, and $(q/p)$-shifting operators. Then, after proving some related properties, and as an application, we considered a generalized $(p,q)$-$\varphi$-difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle.

1. Introduction

As a widely appreciated and exciting field of mathematics, fractional calculus, including fractional and non-fractional differential equations, has developed in recent years. Different types of boundary value problems (BVPs) can be studied with the help of these differential equations, and they can simulate the dynamical behaviors of real phenomena in the world [1]. All these actions and developments in the field of differential equations have been possible due to some important mathematical features of the derivation and integration operators. In other words, in recent years, various mathematicians have tried to define various nonlocal operators with singular or non-singular kernels, which can be used in modeling engineering phenomena, chemical interactions, physical events, and economic issues related to banking and insurance, thereby controlling the spread of deadly viruses in medical science, etc. Among the most widely used operators and some of the studies conducted with them, we can mention the use of the Riemann–Liouville derivative in [2], the Caputo derivative in [2], the Hadamard derivative in [2], the Katugampola derivative in [3], the Hilfer derivative in [4], the Caputo–Fabrizio derivative in [5], and the Atangana–Baleanu derivative in [6].

Quantum calculus, which is referred to as q-calculus in the world of mathematics, created a revolution in the definition of new concepts of derivative and integral operators without relying on the concept of limit, which today can be used on discrete spaces or finite sets. This type of calculus provides the corresponding quantum concepts for most of the previously defined mathematical concepts, in which there is no trace of the concept of limit. In the 18th century, Euler defined some q-formulas in this field, but the main spark of these concepts, which were systematically and precisely struck in the field of mathematics, is related to the works presented by Jackson in 1910 [7,8]. Since then, others have tried to expand on these new q-calculus and to define new q-analog operators of fractional orders, among which mathematicians such as Al-Salam [9], Agarwal [10], and Annaby [11] can be mentioned.

In 2013, Tariboon and Ntouyas [12] introduced an q-shifting operator and used it to define a generalized version of the q-derivative (defined by Jackson) on finite intervals. They applied their q-derivative operator for modeling an impulsive difference equation to show its efficiency. Later, we can see that various papers were published in this field, in which q-derivative operators were used to check the existence and uniqueness of the solutions of q-difference BVPs many times, like in [13,14,15,16,17].

Along with the completion and development of q-calculus, Chakrabarti and Jagannathan [18] used the idea of defining derivative and integral operators without the limit notion, and, this time, they obtained help from two parameters to define new operators. In fact, the new calculus that they founded is known as post-quantum calculus or $(p,q)$-calculus, and, with the assumption of $0<q<p\le 1$, they were able to introduce $(p,q)$-analogs of the above operators. If we pay attention to the basic definitions of $(p,q)$-calculus, we will find that q-calculus is a special case of the $(p,q)$-calculus when the parameter p is equal to one. Bézier surfaces and curves [19], $(p,q)$-approximation methods [20,21], and $(p,q)$-hypergeometric series [22] are a small part of the applications of $(p,q)$-calculus in various mathematical theories. Moreover, in 2020, Soontharanon and Sitthiwirattham [23] conducted a research on fractional $(p,q)$-calculus and defined the fractional $(p,q)$-operators of the two Riemann–Liouville and Caputo types. From that paper onward, other researchers have wanted to continue their studies on mathematical modeling and existence theory based on the existing operators in fractional $(p,q)$-calculus (see [24,25,26,27,28]).

About the main contribution and our novelty in this paper, one can say that we generalized the previously defined $(p,q)$-derivatives and $(p,q)$-integrals to new extended operators with respect to a strictly increasing function $\varphi \left(s\right)$. We call these generalized operators $(p,q)$-$\varphi$-operators. Our main goal was achieved using the definition and properties of three q-shifting, p-shifting, and $(q/p)$-shifting operators (i.e., ${}_{s_0}A_q\left(s\right)$, ${}_{s_0}A_p\left(s\right)$, and ${}_{s_0}A_{(q/p)}\left(s\right)$, respectively) with different orders. Note that these operators are new and cover all special cases. We proved all of the needed properties related to the generalized $(p,q)$-$\varphi$-derivatives and generalized $(p,q)$-$\varphi$-integrals in detail in this paper.

The divisions of this paper are as follows: Everything that will help us to better understand the main content of this paper is collected in Section 2 in a concise and useful way. The new generalized $(p,q)$-$\varphi$-operators and their related properties are presented and proved in Section 3. Section 4 deals with the application of the newly defined $(p,q)$-$\varphi$-operators to prove the existence results of solutions for a generalized $(p,q)$-$\varphi$-difference BVP. Section 5 is related to a numerical example, and Section 6, which is the end of our paper, summarizes the techniques, general results, and subsequent ideas.

2. Preliminaries

Some of the definitions of different versions of the q-derivatives and q-integrals are detailed in this section.

In 1910, Jackson [7,8] defined two operators (known as the Jackson q-operators) with respect to the quantum parameter $0<q<1$ as follows.

Definition 1. 

For the function $\mathsf{ϝ}:[0,L]\to \mathbb{R}$, the q-derivative and q-integral are defined by

$${}_0{}^s\mathbf{D}_q\mathsf{ϝ}\left(s\right)=\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left(qs\right)}{(1-q)s}$$

(1)

and

$$\begin{array}{cc}\hfill {}_0{}^s\mathbf{I}_q\mathsf{ϝ}\left(s\right)& ={\int }_0^s\mathsf{ϝ}\left(u\right){}_0\mathrm{d}_{q}u=(1-q)s\sum _{\mathsf{\ell }=0}^{\infty }{q}^{\mathsf{\ell }}\mathsf{ϝ}(q^{\mathsf{\ell }}s),\hfill \end{array}$$

(2)

respectively.

For $0<q<1$ and $s_0\ge 0$, Tariboon and Ntouyas [12] defined the q-shifting operator ${}_{s_0}A_q\left(s\right)$ (with the start point $s_0$) as

$${}_{s_0}A_q\left(s\right)=qs+(1-q)s_0,s\in \mathbb{R}.$$

It is natural that if $s_0=0$, then ${}_{s_0}A_q\left(s\right)=qs$. Also, ${}_{s_0}A_q\left(1\right)=q$.

Lemma 1 

([29]). For each $0<q<p\le 1$ and $s_0\ge 0$ and $s\in \mathbb{R}$, we have

• ${}_{s_0}A_q\left(s_0\right)=s_0;$
• ${}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)={}_{s_0}A_q^2\left(s\right)={}_{s_0}A_{q^2}\left(s\right);$
• In general, for $\mathsf{\ell }\in \mathbb{N}$, ${}_{s_0}A_q^{\mathsf{\ell }}\left(s\right)={}_{s_0}A_{q^{\mathsf{\ell }}}\left(s\right);$
• ${}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)={}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)={}_{s_0}A_{pq}\left(s\right);$
• ${}_{s_0}A_q^m\left({}_{s_0}A_q^n\left(s\right)\right)={}_{s_0}A_q^n\left({}_{s_0}A_q^m\left(s\right)\right)={}_{s_0}A_q^{m+n}\left(s\right).$

Note that a q-shifting operator can act on a given function like $\mathsf{ϝ}$ as

$$\left({}_{s_0}A_q\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)=\mathsf{ϝ}(qs+(1-q)s_0).$$

The following relation defines the m-times iteration for this operator as

$$\left({}_{s_0}A_q^m\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}\left({}_{s_0}A_q^m\left(s\right)\right)=\mathsf{ϝ}(q^{m}s+(1-q^m)s_0)$$

for all $m\in {\mathbb{Z}}_0$ and ${}_{s_0}A_q^0\mathsf{ϝ}\left(s\right)=\mathsf{ϝ}\left(s\right)$. To investigate the correctness of such a claim, for $m=0$, we have

$$\left({}_{s_0}A_q^0\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}\left({}_{s_0}A_q^0\left(s\right)\right)=\mathsf{ϝ}(q^{0}s+(1-q^0)s_0)=\mathsf{ϝ}\left(s\right).$$

We suppose that $\left({}_{s_0}A_q^k\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}(q^{k}s+(1-q^k)s_0)$ holds for $m=k$. In this case, for $m=k+1$, we can write

$$\begin{array}{cc}\hfill \left({}_{s_0}A_q^{k+1}\mathsf{ϝ}\right)\left(s\right)& ={}_{s_0}A_q\left({}_{s_0}A_q^k\mathsf{ϝ}\left(s\right)\right),\hfill \\ \hfill & ={}_{s_0}A_q\left(\mathsf{ϝ}(q^{k}s+(1-q^k)s_0)\right),\hfill \\ \hfill & =\mathsf{ϝ}\left({}_{s_0}A_q(q^{k}s+(1-q^k)s_0)\right),\hfill \\ \hfill & =\mathsf{ϝ}\left(q(q^{k}s+(1-q^k)s_0)+(1-q)s_0\right),\hfill \\ \hfill & =\mathsf{ϝ}(q^{k+1}s+(1-q^{k+1})s_0).\hfill \end{array}$$

The mathematical induction implies that the m-times iteration of the q-shifting operator ${}_{s_0}A_q$ is in combination if the given function $\mathsf{ϝ}$ is well defined.

Based on [12], the q-shifting-based operators (known as the Tariboon–Ntouyas-type q-operators) were defined in 2013 as follows.

Definition 2 

([12]). For $0<q<1$ and $s_0\ge 0$, the q-shifting-based q-derivative and q-integral operators for the function $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ are given by

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_q\mathsf{ϝ}\left(s\right)& =\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right),}{(1-q)(s-s_0)}\hfill \\ \hfill & =\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}(qs+(1-q)s_0)}{(1-q)(s-s_0)},s\ne s_0,\hfill \end{array}$$

(3)

and

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_q\mathsf{ϝ}\left(s\right)& ={\int }_{s_0}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{q}u,\hfill \\ \hfill & =(1-q)(s-s_0)\sum _{\mathsf{\ell }=0}^{\infty }{q}^{\mathsf{\ell }}\mathsf{ϝ}({}_{s_0}A_q^{\mathsf{\ell }}\left(s\right)),\hfill \\ \hfill & =(1-q)(s-s_0)\sum _{\mathsf{\ell }=0}^{\infty }{q}^{\mathsf{\ell }}\mathsf{ϝ}(q^{\mathsf{\ell }}s+(1-q^{\mathsf{\ell }})s_0),\hfill \end{array}$$

(4)

respectively.

In 2024, Kamsrisuk et al. [30] generalized the q-calculus with the help of another function. In fact, they considered a strictly increasing function like $\varphi :[s_0,L]\to \mathbb{R}$ and presented the following definition of the generalized q-operators with respect to the function $\varphi$ (known as the q-$\varphi$-operators) as follows.

Definition 3 

([30]). For $0<q<1$ and $s_0\ge 0$, the q-derivative and q-integral operators for the function $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ with respect to the function ϕ are given by

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_q^{\varphi }\mathsf{ϝ}\left(s\right)& =\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left(s\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right),}\hfill \\ \hfill & =\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}(qs+(1-q)s_0)}{\varphi \left(s\right)-\varphi (qs+(1-q)s_0)},s\ne s_0,\hfill \end{array}$$

(5)

and

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_q^{\varphi }\mathsf{ϝ}\left(s\right)& ={\int }_{s_0}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_q^{\varphi }u,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_q^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_q^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_q^{\mathsf{\ell }}\left(s\right)\right),\hfill \end{array}$$

(6)

respectively.

In the above definition, it is evident that if $s_0=0$, then

$$\begin{array}{cc}\hfill {}_0{}^s\mathbf{D}_q^{\varphi }\mathsf{ϝ}\left(s\right)& =\frac{\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left(qs\right)}{\varphi \left(s\right)-\varphi \left(qs\right)},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {}_0{}^s\mathbf{I}_q^{\varphi }\mathsf{ϝ}\left(s\right)& ={\int }_0^s\mathsf{ϝ}\left(u\right){}_0\mathrm{d}_q^{\varphi }u=\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left(q^{\mathsf{\ell }}s\right)-\varphi \left(q^{\mathsf{\ell }+1}s\right)\right]\mathsf{ϝ}\left(q^{\mathsf{\ell }}s\right).\hfill \end{array}$$

(8)

In 2018, Sadjang [31] gave a definition of the $(p,q)$-operators (known as the post-quantum operators) for $0<q<p\le 1$ as follows.

Definition 4 

([31]). The $(p,q)$-derivative and $(p,q)$-integral of the function $\mathsf{ϝ}:[0,L]\to \mathbb{R}$ are defined as

$${}_0{}^s\mathbf{D}_{p,q}\mathsf{ϝ}\left(s\right)=\frac{\mathsf{ϝ}\left(ps\right)-\mathsf{ϝ}\left(qs\right)}{(p-q)s},s\ne 0$$

(9)

and

$$\begin{array}{cc}\hfill {}_0{}^s\mathbf{I}_{p,q}\mathsf{ϝ}\left(s\right)& ={\int }_0^s\mathsf{ϝ}\left(u\right){}_0\mathrm{d}_{p,q}u,\hfill \\ \hfill & =(p-q)s\sum _{\mathsf{\ell }=0}^{\infty }\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\mathsf{ϝ}(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s),\hfill \end{array}$$

(10)

respectively.

The q-shifting-based and p-shifting-based operators were defined by Tunç and Göv [32] in 2021 (with the start point $s_0$) as follows.

Definition 5 

([32]). For $0<q<p\le 1$ and $s_0\ge 0$, the $(p,q)$-shifting-based derivative and $(p,q)$-shifting-based integral for the function $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ are given by

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}\mathsf{ϝ}\left(s\right)& =\frac{\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)}{(p-q)(s-s_0)},\hfill \\ \hfill & =\frac{\mathsf{ϝ}(ps+(1-p)s_0)-\mathsf{ϝ}(qs+(1-q)s_0)}{(p-q)(s-s_0)},s\ne s_0,\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_{p,q}\mathsf{ϝ}\left(s\right)& ={\int }_{s_0}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{p,q}u,\hfill \\ \hfill & =(p-q)(s-s_0)\sum _{\mathsf{\ell }=0}^{\infty }\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\mathsf{ϝ}(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}})s_0),\hfill \end{array}$$

(12)

respectively.

3. The Generalized $(p,q)$-$\varphi$-Calculus

As we said before, throughout this paper, $\varphi :[s_0,L]\to \mathbb{R}$ is a strictly increasing function. With respect to this function, we aim to define new generalized derivatives and integrals in this section. To do this, we obtained help from the q-shifting and p-shifting operators with the start point $s_0\ge 0$.

Definition 6. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be a continuous function. The generalized $(p,q)$-ϕ-derivative of ϝ is defined as

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\left(s\right)& =\frac{\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{\mathsf{ϝ}(ps+(1-p)s_0)-\mathsf{ϝ}(qs+(1-q)s_0)}{\varphi (ps+(1-p)s_0)-\varphi (qs+(1-q)s_0)},s\ne s_0.\hfill \end{array}$$

(13)

Moreover,

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\left(s_0\right)=\underset{s\to s_0}{lim}{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\left(s\right).$$

This generalized $(p,q)$-$\varphi$-derivative covers all other previously defined derivatives in q-calculus and $(p,q)$-calculus. The following remark shows this generality.

Remark 1. 

It is notable that

• If $\varphi \left(s\right)=s$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the $(p,q)$-shifting-based derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}$ given by (11) in [32].
• If $\varphi \left(s\right)=s$ and $s_0=0$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the $(p,q)$-derivative ${}_0{}^s\mathbf{D}_{p,q}$ given by (9) in [31].
• If $p=1$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the q-shifting-based derivative ${}_{s_0}{}^s\mathbf{D}_q^{\varphi }$ with respect to the ϕ given by (5) in [30].
• If $p=1$ and $s_0=0$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the q-derivative ${}_0{}^s\mathbf{D}_q^{\varphi }$ with respect to the ϕ given by (7) in [30].
• If $p=1$ and $\varphi \left(s\right)=s$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the q-shifting-based derivative ${}_{s_0}{}^s\mathbf{D}_q$ given by (3) in [12].
• If $p=1$, $\varphi \left(s\right)=s$, and $s_0=0$, then the generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$, given by (13), is reduced to the q-derivative ${}_0{}^s\mathbf{D}_q$ given by (1) in [7].

Lemma 2. 

For $m\in \mathbb{N}$, assume that $\mathsf{ϝ}\left(s\right)={\varphi }^m\left(s\right)$ in (13). Then, we have

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\varphi }^m\left(s\right)=\sum _{\mathsf{\ell }=0}^{m-1}{\varphi }^{m-1-\mathsf{\ell }}(ps+(1-p)s_0){\varphi }^{\mathsf{\ell }}(qs+(1-q)s_0),s\in [s_0,L].$$

(14)

Proof. 

In view of (13), we can write

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }& {\varphi }^m\left(s\right)=\frac{{\varphi }^m\left({}_{s_0}A_q\left(s\right)\right)-{\varphi }^m\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{\left[\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)}[{\varphi }^{m-1}\left({}_{s_0}A_q\left(s\right)\right)+{\varphi }^{m-2}\left({}_{s_0}A_q\left(s\right)\right)\varphi \left({}_{s_0}A_q\left(s\right)\right),\hfill \\ \hfill & +\cdots +\varphi \left({}_{s_0}A_q\left(s\right)\right){\varphi }^{m-2}\left({}_{s_0}A_q\left(s\right)\right)+{\varphi }^{m-1}\left({}_{s_0}A_q\left(s\right)\right)],\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{m-1}{\varphi }^{m-1-\mathsf{\ell }}\left({}_{s_0}A_q\left(s\right)\right){\varphi }^{\mathsf{\ell }}\left({}_{s_0}A_q\left(s\right)\right),\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{m-1}{\varphi }^{m-1-\mathsf{\ell }}(ps+(1-p)s_0){\varphi }^{\mathsf{\ell }}(qs+(1-q)s_0),\hfill \end{array}$$

and the proof is completed. □

Remark 2. 

It is clear that if $p=1$, then (14) is reduced to the following relation:

$${}_{s_0}{}^s\mathbf{D}_q^{\varphi }{\varphi }^m\left(s\right)=\sum _{\mathsf{\ell }=0}^{m-1}{\varphi }^{m-1-\mathsf{\ell }}\left(s\right){\varphi }^{\mathsf{\ell }}(qs+(1-q)s_0),$$

which has been proven in [30] by Kamsrisuk et al. Also, if $p=1$ and $s_0=0$ in (14), then we have

$${}_0{}^s\mathbf{D}_q^{\varphi }{\varphi }^m\left(s\right)=\sum _{\mathsf{\ell }=0}^{m-1}{\varphi }^{m-1-\mathsf{\ell }}\left(s\right){\varphi }^{\mathsf{\ell }}\left(qs\right).$$

Now, we aim to provide an example for computing the generalized $(p,q)$-$\varphi$-derivative with respect to a given function.

Example 1. 

For the two arbitrary numbers $m,k\in \mathbb{N}$ with $m>k$, let $\varphi \left(s\right)={(s-s_0)}^k$. Then, the $(p,q)$-ϕ-derivative of the function $\mathsf{ϝ}\left(s\right)={(s-s_0)}^m$ is computed as

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{{(s-s_0)}^k}{(s-s_0)}^m& =\frac{{({}_{s_0}A_q\left(s\right)-s_0)}^m-{({}_{s_0}A_q\left(s\right)-s_0)}^m}{{({}_{s_0}A_q\left(s\right)-s_0)}^k-{({}_{s_0}A_q\left(s\right)-s_0)}^k},\hfill \\ \hfill & =\frac{{(ps+(1-p)s_0-s_0)}^m-{(qs+(1-q)s_0-s_0)}^m}{{(ps+(1-p)s_0-s_0)}^k-{(qs+(1-q)s_0-s_0)}^k},\hfill \\ \hfill & =\frac{{(ps-p{s}_0)}^m-{(qs-q{s}_0)}^m}{{(ps-p{s}_0)}^k-{(qs-q{s}_0)}^k},\hfill \\ \hfill & =\frac{{\left[p(s-s_0)\right]}^m-{\left[q(s-s_0)\right]}^m}{{\left[p(s-s_0)\right]}^k-{\left[q(s-s_0)\right]}^k},\hfill \\ \hfill & =\frac{p^m{(s-s_0)}^m-q^m{(s-s_0)}^m}{p^k{(s-s_0)}^k-q^k{(s-s_0)}^k},\hfill \\ \hfill & =\frac{(p^m-q^m){(s-s_0)}^m}{(p^k-q^k){(s-s_0)}^k},\hfill \\ \hfill & =\frac{(p^m-q^m)}{(p^k-q^k)}{(s-s_0)}^{m-k}.\hfill \end{array}$$

Remark 3. 

In the above example, if $k=1$, then we obtain

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{(s-s_0)}{(s-s_0)}^m={\left[m\right]}_{p,q}{(s-s_0)}^{m-1}$$

since ${\left[m\right]}_{p,q}=\frac{p^m-q^m}{p-q}$. Also, if $k=1$ and $s_0=0$, then

$${}_0{}^s\mathbf{D}_{p,q}^s{s}^m={\left[m\right]}_{p,q}{s}^{m-1}.$$

Remark 4. 

It is evident that if $k=1$ and $p=1$ in Example 1, then we can obtain the q-ϕ-derivative with respect to the function $\varphi \left(s\right)=s-s_0$ as follows

$${}_{s_0}{}^s\mathbf{D}_q^{(s-s_0)}{(s-s_0)}^m=\frac{(1-q^m)}{(1-q)}{(s-s_0)}^{m-1}={\left[m\right]}_q{(s-s_0)}^{m-1}.$$

Then, we investigated some of the algebraic properties of the generalized $(p,q)$-$\varphi$-derivatives.

Theorem 1. 

(Linear Property) If ${\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2:[s_0,L]\to \mathbb{R}$ are continuous and $c_1,c_2\in \mathbb{R}$ are constants, then

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left[c_1{\mathsf{ϝ}}_1\left(s\right)+c_2{\mathsf{ϝ}}_2\left(s\right)\right]=c_1{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)+c_2{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)$$

for all $s\in [s_0,L]$.

Proof. 

By Definition 6, we have

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }& \left[c_1{\mathsf{ϝ}}_1\left(s\right)+c_2{\mathsf{ϝ}}_2\left(s\right)\right],\hfill \\ \hfill & =\frac{c_1{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)+c_2{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-c_1{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)-c_2{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{c_1\left[{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)\right]+c_2\left[{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\right]}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{c_1\left[{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)\right]}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)}+\frac{c_2\left[{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\right]}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =c_1{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)+c_2{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right).\hfill \end{array}$$

This completes the proof. □

The next theorem proves how the generalized $(p,q)$-$\varphi$-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$ acts on the product of two functions.

Theorem 2. 

For two continuous functions ${\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2:[s_0,L]\to \mathbb{R}$, the generalized $(p,q)$-ϕ-derivative of the product function ${\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2$ is given by

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right)& ={\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)+{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right),\hfill \\ \hfill & ={\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)+{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)\hfill \end{array}$$

for all $s\in [s_0,L]$.

Proof. 

For each $s\in [s_0,L]$ and by Definition 6, we have

$$\begin{array}{cc}\hfill & {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right)=\frac{({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left({}_{s_0}A_q\left(s\right)\right)-({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)+{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)\left[{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\right]+{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\left[{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)\right]}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & ={\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)+{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right).\hfill \end{array}$$

The second case is similarly proved. □

For a quotient function, the generalized $(p,q)$-$\varphi$-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$ acted as follows.

Theorem 3. 

For two continuous functions ${\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2:[s_0,L]\to \mathbb{R}$, the generalized $(p,q)$-ϕ-derivative of the quotient function ${\mathsf{ϝ}}_1/{\mathsf{ϝ}}_2$ was given by

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left(\frac{{\mathsf{ϝ}}_1}{{\mathsf{ϝ}}_2}\right)\left(s\right)=\frac{{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)}{{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)},$$

so that ${\mathsf{ϝ}}_2\left(s\right)\ne 0$ for all $s\in [s_0,L]$.

Proof. 

By Definition 6, for each $s\in [s_0,L]$, we have

$$\begin{array}{c}{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left(\frac{{\mathsf{ϝ}}_1}{{\mathsf{ϝ}}_2}\right)\left(s\right)=\frac{\left(\frac{{\mathsf{ϝ}}_1}{{\mathsf{ϝ}}_2}\right)\left({}_{s_0}A_q\left(s\right)\right)-\left(\frac{{\mathsf{ϝ}}_1}{{\mathsf{ϝ}}_2}\right)\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ =\frac{{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)+{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)}{{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\left[\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]},\hfill \\ =\frac{{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right)\left[{\mathsf{ϝ}}_1\left({}_{s_0}A_p\left(s\right)\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right)\right]-{\mathsf{ϝ}}_1\left({}_{s_0}A_p\left(s\right)\right)\left[{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\right]}{{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)\left[\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]},\hfill \\ =\frac{{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)-{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)}{{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right){\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(s\right)\right)},\hfill \end{array}$$

and this completes the proof. □

Sometimes, we may need to compute the generalized $(p,q)$-$\varphi$-derivative of the higher order (greater than one). For instance, let us obtain help from Definition 6 and compute ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }$ of the second order as follows:

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }\mathsf{ϝ}\left(s\right)& ={}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right),\hfill \\ \hfill & =\frac{{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\left({}_{s_0}A_p\left(s\right)\right)-{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{\frac{\mathsf{ϝ}\left({}_{s_0}A_p\left({}_{s_0}A_p\left(s\right)\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_p\left({}_{s_0}A_q\left(s\right)\right)\right)}{\varphi \left({}_{s_0}A_p\left({}_{s_0}A_p\left(s\right)\right)\right)-\varphi \left({}_{s_0}A_p\left({}_{s_0}A_q\left(s\right)\right)\right)}-\frac{\mathsf{ϝ}\left({}_{s_0}A_q\left({}_{s_0}A_p\left(s\right)\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)\right)}{\varphi \left({}_{s_0}A_q\left({}_{s_0}A_p\left(s\right)\right)\right)-\varphi \left({}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)\right)}}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)}.\hfill \end{array}$$

(15)

By Lemma 1, we know that

$${}_{s_0}A_q\left({}_{s_0}A_q\left(s\right)\right)={}_{s_0}A_q^2\left(s\right)={}_{s_0}A_{q^2}\left(s\right)=q^{2}s+(1-q^2)s_0,$$

$${}_{s_0}A_p\left({}_{s_0}A_p\left(s\right)\right)={}_{s_0}A_p^2\left(s\right)={}_{s_0}A_{p^2}\left(s\right)=p^{2}s+(1-p^2)s_0,$$

and

$${}_{s_0}A_q\left({}_{s_0}A_p\left(s\right)\right)={}_{s_0}A_p\left({}_{s_0}A_q\left(s\right)\right)={}_{s_0}A_{pq}\left(s\right)=pqs+(1-pq)s_0.$$

Thus, by continuing (15), we obtain

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }\mathsf{ϝ}\left(s\right)& =\frac{\frac{\mathsf{ϝ}\left({}_{s_0}A_{p^2}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right)}{\varphi \left({}_{s_0}A_{p^2}\left(s\right)\right)-\varphi \left({}_{s_0}A_{pq}\left(s\right)\right)}-\frac{\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{q^2}\left(s\right)\right)}{\varphi \left({}_{s_0}A_{pq}\left(s\right)\right)-\varphi \left({}_{s_0}A_{q^2}\left(s\right)\right)}}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{\left[\mathsf{ϝ}\left({}_{s_0}A_{p^2}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right)\right]{\Psi }_1\left(s\right)-\left[\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{q^2}\left(s\right)\right)\right]{\Psi }_2\left(s\right)}{{\Psi }_1\left(s\right){\Psi }_2\left(s\right)\left[\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]},\hfill \\ \hfill & =\frac{\mathsf{ϝ}\left({}_{s_0}A_{p^2}\left(s\right)\right){\Psi }_1\left(s\right)+\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right){\Psi }_3\left(s\right)+\mathsf{ϝ}\left({}_{s_0}A_{q^2}\left(s\right)\right){\Psi }_2\left(s\right)}{{\Psi }_1\left(s\right){\Psi }_2\left(s\right)\left[\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]},\hfill \end{array}$$

where,

$${\Psi }_1\left(s\right)=\left[\varphi \left({}_{s_0}A_{pq}\left(s\right)\right)-\varphi \left({}_{s_0}A_{q^2}\left(s\right)\right)\right],$$

$${\Psi }_2\left(s\right)=\left[\varphi \left({}_{s_0}A_{p^2}\left(s\right)\right)-\varphi \left({}_{s_0}A_{pq}\left(s\right)\right)\right],$$

and

$${\Psi }_3\left(s\right)=\left[\varphi \left({}_{s_0}A_{q^2}\left(s\right)\right)-\varphi \left({}_{s_0}A_{p^2}\left(s\right)\right)\right]$$

for $s\ne s_0$, and ${}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }\mathsf{ϝ}\left(s_0\right)={\displaystyle \underset{s\to s_0}{lim}}{}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }\mathsf{ϝ}\left(s\right)$.

Remark 5. 

One can easily seen that if $\varphi \left(s\right)=s$, then the second-order $(p,q)$-shifting-based $(p,q)$-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^2$ (proven by Tunç and Göv in [32]) is obtained as

$${}_{s_0}{}^s\mathbf{D}_{p,q}^2\mathsf{ϝ}\left(s\right)=\frac{q\mathsf{ϝ}\left({}_{s_0}A_{p^2}\left(s\right)\right)-(p+q)\mathsf{ϝ}\left({}_{s_0}A_{pq}\left(s\right)\right)+p\mathsf{ϝ}\left({}_{s_0}A_{q^2}\left(s\right)\right)}{pq{(p-q)}^2{(s-s_0)}^2},s\ne s_0.$$

Based on the above computations and results, we can present a definition of the second-order generalized $(p,q)$-$\varphi$-derivatives as follows.

Definition 7. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be continuous and ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}$ be $(p,q)$-ϕ-differentiable on $[s_0,L]$. Then, the second-order generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }$ is defined by

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{2,\varphi }\mathsf{ϝ}\left(s\right)={}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right)$$

on $[s_0,L]$. Via this method, the m-th order generalized $(p,q)$-ϕ-derivative ${}_{s_0}{}^s\mathbf{D}_{p,q}^{m,\varphi }$ is defined by

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{m,\varphi }\mathsf{ϝ}\left(s\right)={}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{m-1,\varphi }\mathsf{ϝ}\right)\left(s\right),m\in \mathbb{N}.$$

By considering the strictly increasing function $\varphi$ as above, defining the generalized $(p,q)$-$\varphi$-integral was our next step in this paper. To achieve this, we considered the $(q/p)$-shifting operator ${}_{s_0}A_{\frac{q}{p}}$ acting on the function $\mathfrak{F}$ (according to the previous rule stated in Section 2), which is the generalized $(p,q)$-$\varphi$-anti-derivative of $\mathsf{ϝ}$:

$${}_{s_0}A_{\frac{q}{p}}\mathfrak{F}\left(s\right)=\mathfrak{F}\left(\frac{q}{p}s+(1-\frac{q}{p})s_0\right),0<\frac{q}{p}\le 1.$$

By the property of the m-times iteration for the operator ${}_{s_0}A_{\frac{q}{p}}$, we know that

$${}_{s_0}A_{\frac{q}{p}}^m\mathfrak{F}\left(s\right)=\mathfrak{F}\left(\frac{q^m}{p^m}s+(1-\frac{q^m}{p^m})s_0\right),m\in \mathbb{N},$$

(16)

and ${}_{s_0}A_{\frac{q}{p}}^0\mathfrak{F}\left(s\right)=\mathfrak{F}\left(s\right)$.

By Definition 6, one may write

$$\begin{array}{cc}\hfill \mathsf{ϝ}\left(s\right)& =\frac{\mathfrak{F}\left({}_{s_0}A_p\left(s\right)\right)-\mathfrak{F}\left({}_{s_0}A_q\left(s\right)\right)}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)},\hfill \\ \hfill & =\frac{\mathfrak{F}(ps+(1-p)s_0)-\mathfrak{F}(qs+(1-q)s_0)}{\varphi (ps+(1-p)s_0)-\varphi (qs+(1-q)s_0)}.\hfill \end{array}$$

We use the change in the variable for $ps+(1-p)s_0=r$. Then,

$$\begin{array}{cc}\hfill \mathsf{ϝ}\left(\frac{r-(1-p)s_0}{p}\right)& =\frac{\mathfrak{F}\left(r\right)-\mathfrak{F}\left(\frac{q}{p}r+(1-\frac{q}{p})s_0\right)}{\varphi \left(r\right)-\varphi \left(\frac{q}{p}r+(1-\frac{q}{p})s_0\right)},\hfill \\ \hfill & =\frac{\mathfrak{F}\left(r\right)-\mathfrak{F}({}_{s_0}A_{\frac{q}{p}}\left(r\right))}{\varphi \left(r\right)-\varphi ({}_{s_0}A_{\frac{q}{p}}\left(r\right))},\hfill \\ \hfill & =\frac{\mathfrak{F}\left(r\right)-{}_{s_0}A_{\frac{q}{p}}\mathfrak{F}\left(r\right)}{\varphi \left(r\right)-\varphi ({}_{s_0}A_{\frac{q}{p}}\left(r\right))},\hfill \\ \hfill & =\frac{(1-{}_{s_0}A_{\frac{q}{p}})}{\varphi \left(r\right)-\varphi ({}_{s_0}A_{\frac{q}{p}}\left(r\right))}\mathfrak{F}\left(r\right).\hfill \end{array}$$

Then, the generalized $(p,q)$-$\varphi$-anti-derivative can be formulated as

$$\mathfrak{F}\left(r\right)=\frac{1}{1-{}_{s_0}A_{\frac{q}{p}}}\left[\varphi \left(r\right)-\varphi ({}_{s_0}A_{\frac{q}{p}}\left(r\right))\right]\mathsf{ϝ}\left(\frac{r-(1-p)s_0}{p}\right).$$

In this step, if we applied the expansion of the geometric series on (16), then it gives

$$\begin{array}{cc}\hfill \mathfrak{F}\left(r\right)& =\sum _{m=0}^{\infty }{}_{s_0}A_{\frac{q}{p}}^m\left[\varphi \left(r\right)-\varphi ({}_{s_0}A_{\frac{q}{p}}\left(r\right))\right]\mathsf{ϝ}\left(\frac{r-(1-p)s_0}{p}\right),\hfill \\ \hfill & =\sum _{m=0}^{\infty }{}_{s_0}A_{\frac{q}{p}}^m\left[\varphi \left(r\right)-\varphi \left(\frac{q}{p}r+(1-\frac{q}{p}\right)s_0)\right]\mathsf{ϝ}\left(\frac{r-(1-p)s_0}{p}\right),\hfill \\ \hfill & =\sum _{m=0}^{\infty }\left[\varphi (\frac{q^m}{p^m}r+\left(1-\frac{q^m}{p^m}\right)s_0)-\varphi \left(\frac{q^{m+1}}{p^{m+1}}r+\frac{q^m}{p^m}(1-\frac{q}{p}\right)s_0+\left(1-\frac{q^m}{p^m}\right)s_0)\right],\hfill \\ \hfill & \times \mathsf{ϝ}\left(\frac{1}{p}\left(\frac{q^m}{p^m}r+\left(1-\frac{q^m}{p^m}\right)s_0-(1-p)s_0\right)\right),\hfill \\ \hfill & =\sum _{m=0}^{\infty }[\varphi \left(\frac{q^m}{p^m}r+\left(1-\frac{q^m}{p^m}\right)s_0)-\varphi \left(\frac{q^{m+1}}{p^{m+1}}r+\left(1-\frac{q^{m+1}}{p^{m+1}}\right)s_0\right)\right],\hfill \\ \hfill & \times \mathsf{ϝ}\left(\frac{q^m}{p^{m+1}}r+\left(1-\frac{q^m}{p^{m+1}}\right)s_0\right)\hfill \end{array}$$

provided that the right-hand side of the above series is convergent.

In this position, we can present our definition.

Definition 8. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be continuous. For each $0<q<p\le 1$ and $s\in [s_0,L]$, we define the generalized $(p,q)$-ϕ-integral of the function ϝ as follows

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\mathsf{ϝ}\left(s\right)& ={\int }_{s_0}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi ({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right))-\varphi ({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right))\right]\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}\right)s_0\right)-\varphi \left(\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right],\hfill \\ \hfill & \times \mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \end{array}$$

(17)

so that the right-hand side of the above series is convergent.

Remark 6. 

It is notable that

• If $\varphi \left(s\right)=s$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the $(p,q)$-shifting-based integral ${}_{s_0}{}^s\mathbf{I}_{p,q}$ given by (12) in [32].
• If $\varphi \left(s\right)=s$ and $s_0=0$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the $(p,q)$-integral ${}_0{}^s\mathbf{I}_{p,q}$ given by (10) in [31].
• If $p=1$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the q-shifting-based derivative ${}_{s_0}{}^s\mathbf{I}_q^{\varphi }$ with respect to the ϕ given by (6) in [30].
• If $p=1$ and $s_0=0$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the q-integral ${}_0{}^s\mathbf{I}_q^{\varphi }$ with respect to the ϕ given by (8) in [30].
• If $p=1$ and $\varphi \left(s\right)=s$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the q-shifting-based integral ${}_{s_0}{}^s\mathbf{I}_q$ given by (4) in [12].
• If $p=1$, $\varphi \left(s\right)=s$, and $s_0=0$, then the generalized $(p,q)$-ϕ-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$, given by (17), is reduced to the q-integral ${}_0{}^s\mathbf{I}_q$ given by (2) in [7].

If $s^{*}\in (s_0,L)$, we can define the definite generalized $(p,q)$-$\varphi$-integral as

$$\begin{array}{cc}\hfill {\int }_{s^{*}}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u& ={\int }_{s_0}^s\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u-{\int }_{s_0}^{s^{*}}\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & -\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s^{*}\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s^{*}\right)\right)\right]\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}{s}^{*}+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right).\hfill \end{array}$$

In the next example, we computed a generalized $(p,q)$-$\varphi$-integral for a given arbitrary function.

Example 2. 

For two arbitrary numbers $m,k\ge 0$, let $\mathsf{ϝ}\left(s\right)={(s-s_0)}^k$ and $\varphi \left(s\right)={(s-s_0)}^m$. Then, we have

$$\begin{array}{cc}\hfill {\int }_{s_0}^s{(u-s_0)}^k{}_{s_0}\mathrm{d}_{p,q}^{{(u-s_0)}^m}u& =\sum _{\mathsf{\ell }=0}^{\infty }\left[{\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}\right)s_0-s_0\right)}^m-{\left(\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}\right)s_0-s_0\right)}^m\right],\hfill \\ \hfill & \times {\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0-s_0\right)}^k,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[{\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}s-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}{s}_0\right)}^m-{\left(\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}s-\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}{s}_0\right)}^m\right]{\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}{s}_0\right)}^k,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[{\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}(s-s_0)\right)}^m-{\left(\frac{q^{\mathsf{\ell }+1}}{p^{\mathsf{\ell }+1}}(s-s_0)\right)}^m\right]{\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}(s-s_0)\right)}^k,\hfill \\ \hfill & ={(s-s_0)}^{m+k}\sum _{\mathsf{\ell }=0}^{\infty }\left[{\left(\frac{q^m}{p^m}\right)}^{\mathsf{\ell }}-{\left(\frac{q^m}{p^m}\right)}^{\mathsf{\ell }+1}\right]{\left(\frac{q^k}{p^k}\right)}^{\mathsf{\ell }}\frac{1}{p^k},\hfill \\ \hfill & ={(s-s_0)}^{m+k}\frac{1}{p^k}\left[\sum _{\mathsf{\ell }=0}^{\infty }{\left(\frac{q^{m+k}}{p^{m+k}}\right)}^{\mathsf{\ell }}-\frac{q^m}{p^m}\sum _{\mathsf{\ell }=0}^{\infty }{\left(\frac{q^{m+k}}{p^{m+k}}\right)}^{\mathsf{\ell }}\right],\hfill \\ \hfill & ={(s-s_0)}^{m+k}\frac{1}{p^k}\left[\frac{1}{1-\frac{q^{m+k}}{p^{m+k}}}-\frac{q^m}{p^m}\frac{1}{1-\frac{q^{m+k}}{p^{m+k}}}\right],\hfill \\ \hfill & ={(s-s_0)}^{m+k}\frac{1}{p^k}\left[\left(1-{\left(\frac{q}{p}\right)}^m\right)\left(\frac{1}{1-{\left(\frac{q}{p}\right)}^{m+k}}\right)\right].\hfill \end{array}$$

In the above example, we have the following special cases:

• If $m=k=1$, then

  $${\int }_{s_0}^s(u-s_0){}_{s_0}\mathrm{d}_{p,q}^{u-s_0}u=\frac{{(s-s_0)}^2}{p}\left[\frac{1-\frac{q}{p}}{1-\frac{q^2}{p^2}}\right]=\frac{{(s-s_0)}^2}{p+q}=\frac{{(s-s_0)}^2}{{\left[2\right]}_{p,q}}.$$
• If $m=1$ and $k=0$, then

  $${\int }_{s_0}^s{}_{s_0}\mathrm{d}_{p,q}^{u-s_0}u=s-s_0.$$
• If $m=0$ and $k=1$, then

  $${\int }_{s_0}^s(u-s_0){}_{s_0}\mathrm{d}_{p,q}^{1}u=0.$$
• If $m=k=p=1$, then

  $${\int }_{s_0}^s(u-s_0){}_{s_0}\mathrm{d}_q^{u-s_0}u={(s-s_0)}^2\left[\frac{1-q}{1-q^2}\right]=\frac{{(s-s_0)}^2}{1+q}=\frac{{(s-s_0)}^2}{{\left[2\right]}_q}.$$

Remark 7. 

By Definition 8, one can easily seen that if $\mathsf{ϝ}\left(s\right)=1$, then we have

$${\int }_{s_0}^{s}1{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u=\varphi \left(s\right)-\varphi \left(s_0\right).$$

We now present some fundamental theorems on the properties of the generalized $(p,q)$-$\varphi$-operators.

Theorem 4. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be continuous. For all $s\in [s_0,L]$, we have

$${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}\left(s\right).$$

Proof. 

If we use Definitions 6 and 8 for the generalized $(p,q)$-$\varphi$-integral and $(p,q)$-$\varphi$-derivative, respectively, then we have

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }& \left({}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right),\hfill \\ \hfill & ={}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\left[\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right],\hfill \\ \hfill & =\frac{1}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)}[\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_p\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)\right)-\varphi \left({}_{s_0}A_p\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right)\right],\hfill \\ \hfill & \times \mathsf{ϝ}\left({}_{s_0}A_p\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),\hfill \\ \hfill & -\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_q\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)\right)-\varphi \left({}_{s_0}A_q\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_q(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}})s_0)\right)],\hfill \\ \hfill & =\frac{1}{\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)}[\left[\varphi \left({}_{s_0}A_p\left(s\right)\right)-\varphi \left({}_{s_0}A_q\left(s\right)\right)\right]\mathsf{ϝ}\left(s\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi ({}_{s_0}A_{\frac{q^2}{p}}\left(s\right))\right]\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q^2}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^3}{p^2}}\left(s\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right),\hfill \\ \hfill & +\cdots \cdots \cdots \hfill \\ \hfill & -\left[\varphi \left({}_{s_0}A_q\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^2}{p}}\left(s\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right),\hfill \\ \hfill & -\left[\varphi \left({}_{s_0}A_{\frac{q^2}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^3}{p^2}}\left(s\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right),\hfill \\ \hfill & -\left[\varphi \left({}_{s_0}A_{\frac{q^3}{p^2}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^4}{p^3}}\left(s\right)\right)\right]\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^3\left(s\right)\right)-\cdots \cdots \cdots ],\hfill \\ \hfill & =\mathsf{ϝ}\left(s\right),\hfill \end{array}$$

which proves the theorem. □

Theorem 5. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be continuous. For all $s\in (s_0,L]$, we have

$${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right)=\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left(s_0\right).$$

Proof. 

If we use Definitions 6 and 8 for the generalized $(p,q)$-$\varphi$-derivative and $(p,q)$-$\varphi$-integral, respectively, then we have

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ}\right)\left(s\right)& ={\int }_{s_0}^s{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathsf{ϝ})\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & ={\int }_{s_0}^s\frac{\mathsf{ϝ}\left({}_{s_0}A_p\left(u\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_q\left(u\right)\right)}{\varphi \left({}_{s_0}A_p\left(u\right)\right)-\varphi \left({}_{s_0}A_q\left(u\right)\right)}{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right],\hfill \\ \hfill & \times \frac{\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\left({}_{s_0}A_p\left(s\right)\right)+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)-\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\left({}_{s_0}A_q\left(s\right)\right)+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)}{\varphi \left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\left({}_{s_0}A_p\left(s\right)\right)+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)-\varphi \left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\left({}_{s_0}A_q\left(s\right)\right)+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)},\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\frac{\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)}{\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)},\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left(\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right),\hfill \\ \hfill & =\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)+\left(\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)\right),\hfill \\ \hfill & +\left(\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)-\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p}}^3\left(s\right)\right)\right)+\cdots ,\hfill \\ \hfill & =\mathsf{ϝ}\left(s\right)-\mathsf{ϝ}\left(s_0\right),\hfill \end{array}$$

so that $\underset{\mathsf{\ell }\to \infty }{lim}\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }}}\right)s_0\right)=\mathsf{ϝ}\left(s_0\right)$. This completes the proof. □

The linear property is one of the most important specifications of the generalized $(p,q)$-$\varphi$-integral, which is proved in the following.

Theorem 6. 

(Linear Property) If ${\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2:[s_0,L]\to \mathbb{R}$ are continuous and $c_1,c_2\in \mathbb{R}$ are constants, then

$${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\left[c_1{\mathsf{ϝ}}_1\left(s\right)+c_2{\mathsf{ϝ}}_2\left(s\right)\right]=c_1{}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)+c_2{}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)$$

for all $s\in [s_0,L]$.

Proof. 

By Definition 8, we have

$$\begin{array}{cc}\hfill {}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\left[c_1{\mathsf{ϝ}}_1\left(s\right)+c_2{\mathsf{ϝ}}_2\left(s\right)\right]& ={\int }_{s_0}^s\left(c_1{\mathsf{ϝ}}_1\left(u\right)+c_2{\mathsf{ϝ}}_2\left(u\right)\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right],\hfill \\ \hfill & \times \left(c_1{\mathsf{ϝ}}_1\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)+c_2{\mathsf{ϝ}}_2\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =c_1\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]{\mathsf{ϝ}}_1\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & +c_2\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]{\mathsf{ϝ}}_2\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & =c_1{\int }_{s_0}^s{\mathsf{ϝ}}_1\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+c_2{\int }_{s_0}^s{\mathsf{ϝ}}_2\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =c_1{}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right)+c_2{}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right),\hfill \end{array}$$

and the proof is completed. □

The next theorem proves the $(p,q)$-$\varphi$-integration by parts for the generalized $(p,q)$-$\varphi$-integrals.

Theorem 7. 

Let ${\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2:[s_0,L]\to \mathbb{R}$ be continuous. Then,

(i) 

${\int }_{s_0}^s{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u=({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right){|}_{s_0}^s-{\int }_{s_0}^s{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u$;

(ii) 

${\int }_{s_0}^s{\mathsf{ϝ}}_1\left({}_{s_0}A_p\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u=({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right){|}_{s_0}^s-{\int }_{s_0}^s{\mathsf{ϝ}}_2\left({}_{s_0}A_q\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u$.

Proof. 

(i) By Theorem 2, one can write

$${\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(s\right)={}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right)-{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(s\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(s\right).$$

By $(p,q)$-$\varphi$-integrating on $[s_0,s]$ and by Theorems 5 and 6, we obtain

$$\begin{array}{cc}\hfill {\int }_{s_0}^s{\mathsf{ϝ}}_1\left({}_{s_0}A_q\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_2\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u& ={\int }_{s_0}^s{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & -{\int }_{s_0}^s{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & =({\mathsf{ϝ}}_1·{\mathsf{ϝ}}_2)\left(s\right){|}_{s_0}^s-{\int }_{s_0}^s{\mathsf{ϝ}}_2\left({}_{s_0}A_p\left(u\right)\right){}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }{\mathsf{ϝ}}_1\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u.\hfill \end{array}$$

The second one (Property (ii)) is similarly proved. □

The next theorem is related to the double $(p,q)$-$\varphi$-integration, and this shows that every double $(p,q)$-$\varphi$-integral can be transformed into a single $(p,q)$-$\varphi$-integral.

Theorem 8. 

Let $\mathsf{ϝ}:[s_0,L]\to \mathbb{R}$ be continuous. Then,

$${\int }_{s_0}^s{\int }_{s_0}^u\mathsf{ϝ}\left(\tau \right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }\tau {}_{s_0}\mathrm{d}_{p,q}^{\varphi }u={\int }_{s_0}^s\left[\varphi \left(u\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(u\right)\right)\right]·\mathsf{ϝ}\left(\frac{1}{p}u+\left(1-\frac{1}{p}\right)s_0\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u.$$

Proof. 

Definition 8 implies that

$$\begin{array}{cc}\hfill {\int }_{s_0}^s{\int }_{s_0}^u& \mathsf{ϝ}\left(\tau \right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }\tau {}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & ={\int }_{s_0}^s\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(u\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(u\right)\right)\right]·\mathsf{ϝ}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}u+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{j=0}^{\infty }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^j\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{j+1}\left(s\right)\right)\right][\varphi \left(\frac{q^j}{p^{j+1}}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(t\right)\right)+\left(1-\frac{q^j}{p^{j+1}}\right)s_0\right),\hfill \\ \hfill & -\varphi \left(\frac{q^j}{p^{j+1}}\left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(t\right)\right)+\left(1-\frac{q^j}{p^{j+1}}\right)s_0\right)]·\mathsf{ϝ}\left(\frac{q^j}{p^{j+1}}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)+\left(1-\frac{q^j}{p^{j+1}}\right)s_0\right),\hfill \\ \hfill & =\sum _{j=0}^{\infty }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^j\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{j+1}\left(s\right)\right)\right][\varphi \left(\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+1}}s+\left(1-\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & -\varphi \left(\frac{q^{j+\mathsf{\ell }+1}}{p^{j+\mathsf{\ell }+2}}s+\left(1-\frac{q^{j+\mathsf{\ell }+1}}{p^{j+\mathsf{\ell }+2}}\right)s_0\right)]·\mathsf{ϝ}\left(\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+2}}s+\left(1-\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+2}}\right)s_0\right),\hfill \\ \hfill & =\sum _{j=0}^{\infty }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^j\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{j+1}\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+1}}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^{j+\mathsf{\ell }+1}}{p^{j+\mathsf{\ell }+2}}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^{j+\mathsf{\ell }}}{p^{j+\mathsf{\ell }+2}}}\left(s\right)\right),\hfill \\ \hfill & =\left[\varphi \left(s\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{1}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p^2}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{1}{p^2}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left(s\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q}{p^2}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p^3}}\left(s\right)\right)+\cdots \cdots ,\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q}{p^2}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p^3}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^3}{p^4}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^2}{p^4}}\left(s\right)\right)+\cdots \cdots ,\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^3\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^3}{p^4}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^2}{p^4}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^3\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^3}{p^4}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^4}{p^5}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^3}{p^5}}\left(s\right)\right)+\cdots \cdots ,\hfill \\ \hfill & =\left[\varphi \left(s\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{1}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p^2}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{1}{p^2}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q}{p^2}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q}{p^3}}\left(s\right)\right),\hfill \\ \hfill & +\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^2\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^3\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^2}{p^3}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^3}{p^4}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^2}{p^4}}\left(s\right)\right),\hfill \\ \hfill & =\sum _{i=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^i\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{i+1}\left(s\right)\right)\right]\left[\varphi \left({}_{s_0}A_{\frac{q^i}{p^{i+1}}}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q^{i+1}}{p^{i+2}}}\left(s\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{q^i}{p^{i+2}}}\left(s\right)\right),\hfill \\ \hfill & =\sum _{i=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^i\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{i+1}\left(s\right)\right)\right]\left[\varphi \left(\frac{q^i}{p^{i+1}}s+\left(1-\frac{q^i}{p^{i+1}}\right)s_0\right)-\varphi \left(\frac{q^{i+1}}{p^{i+2}}s+\left(1-\frac{q^{i+1}}{p^{i+2}}\right)s_0\right)\right],\hfill \\ \hfill & \times \mathsf{ϝ}\left(\frac{q^i}{p^{i+2}}s+\left(1-\frac{q^i}{p^{i+2}}\right)s_0\right),\hfill \\ \hfill & ={\int }_{s_0}^s\left[\varphi \left(u\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(u\right)\right)\right]·\mathsf{ϝ}\left({}_{s_0}A_{\frac{1}{p}}\left(u\right)\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & ={\int }_{s_0}^s\left[\varphi \left(u\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}\left(u\right)\right)\right]·\mathsf{ϝ}\left(\frac{1}{p}u+\left(1-\frac{1}{p}\right)s_0\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \end{array}$$

and the proof is completed. □

Remark 8. 

It is clear that if $p=1$, then we have the following generalized double q-ϕ-integration, which is proved in [30] as

$${\int }_{s_0}^s{\int }_{s_0}^u\mathsf{ϝ}\left(\tau \right){}_{s_0}\mathrm{d}_q^{\varphi }\tau {}_{s_0}\mathrm{d}_q^{\varphi }u={\int }_{s_0}^s\left[\varphi \left(u\right)-\varphi \left({}_{s_0}A_q\left(u\right)\right)\right]·\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_q^{\varphi }u.$$

Remark 9. 

If $\varphi \left(s\right)=s$ and $p=1$, then we have the standard double q-integration as

$${\int }_{s_0}^s{\int }_{s_0}^u\mathsf{ϝ}\left(\tau \right){}_{s_0}\mathrm{d}_q\tau {}_{s_0}\mathrm{d}_{q}u=(1-q){\int }_{s_0}^s(u-s_0)\mathsf{ϝ}\left(u\right){}_{s_0}\mathrm{d}_{q}u.$$

4. Application to a $(p,q)$-$\varphi$-Difference BVP

As shown in this section, we intended to test an application of the generalized $(p,q)$-$\varphi$-operators for studying the existence theory in relation to the solutions of a generalized $(p,q)$-$\varphi$-difference boundary value problem. To achieve this, we considered the following generalized $(p,q)$-$\varphi$-difference problem given as

$$\left\{\begin{array}{c}{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathfrak{f}\left(s\right)=\mathfrak{h}(s,\mathfrak{f}\left(s\right)),\hfill \\ \mathfrak{f}\left(s_0\right)=\lambda \mathfrak{f}\left(L\right),s\in [s_0,L],\hfill \end{array}\right.$$

(18)

so that $\lambda \in \mathbb{R}$ with $\lambda <1$, $0<q<p\le 1$, $L>s_0$, $s_0\ge 0$, $\varphi$ is a strictly increasing function on $[s_0,L]$, and $\mathfrak{h}\in C([s_0,L]\times \mathbb{R},\mathbb{R})$ is nonlinear. Note that if we set $\lambda =-1$, then the above problem is reduced to an anti-periodic problem.

The first step in our study was that we transform the above generalized $(p,q)$-$\varphi$-difference problem (18) into a generalized $(p,q)$-$\varphi$-integral equation by assuming its linear variant.

Lemma 3. 

Let $\mathcal{T}\in C([s_0,L]\times \mathbb{R},\mathbb{R})$ and $\lambda \in {\mathbb{R}}^{+}$ with $\lambda <1$. Then, the generalized linear $(p,q)$-ϕ-difference problem

$$\left\{\begin{array}{c}{}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathfrak{f}\left(s\right)=\mathcal{T}\left(s\right),\hfill \\ \mathfrak{f}\left(s_0\right)=\lambda \mathfrak{f}\left(L\right),s\in [s_0,L],\hfill \end{array}\right.$$

(19)

has a unique solution in a form as

$$\begin{array}{cc}\hfill \mathfrak{f}\left(s\right)& =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathcal{T}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & +\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right]\mathcal{T}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \end{array}$$

(20)

and vice versa.

Proof. 

If the generalized $(p,q)$-$\varphi$-integral ${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }$ acts on both sides of the generalized $(p,q)$-$\varphi$-difference equation ${}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathfrak{f}\left(s\right)=\mathcal{T}\left(s\right)$ in (19), then we have

$${}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\left({}_{s_0}{}^s\mathbf{D}_{p,q}^{\varphi }\mathfrak{f}\right)\left(s\right)=\left({}_{s_0}{}^s\mathbf{I}_{p,q}^{\varphi }\mathcal{T}\right)\left(s\right),$$

and so

$$\mathfrak{f}\left(s\right)=\mathfrak{f}\left(s_0\right)+{\int }_{s_0}^s\mathcal{T}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u.$$

(21)

The given boundary condition $\mathfrak{f}\left(s_0\right)=\lambda \mathfrak{f}\left(L\right)$ gives

$$\mathfrak{f}\left(s_0\right)=\frac{\lambda }{1-\lambda }{\int }_{s_0}^L\mathcal{T}\left(u\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u.$$

This means that

$$\mathfrak{f}\left(s_0\right)=\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right]\mathcal{T}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right).$$

(22)

If we put (22) into (21), then we have

$$\begin{array}{cc}\hfill \mathfrak{f}\left(s\right)& =\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathcal{T}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right),\hfill \\ \hfill & +\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right]\mathcal{T}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right).\hfill \end{array}$$

Thus, $\mathfrak{f}\left(s\right)$ satisfies the generalized $(p,q)$-$\varphi$-integral Equation (20). For the converse, with the help of Theorem 4 and by direct calculations, one can also show that $\mathfrak{f}\left(s\right)$ (as a solution of (20)) satisfies the generalized $(p,q)$-$\varphi$-difference problem (19). The proof is completed in this step. □

Now, we need a Banach space (containing all the continuous functions) to prove the related theorems on the existence theory of the solutions. For this reason, we defined such a space by

$$\widehat{F}:=C([s_0,L],\mathbb{R}):=\left\{\mathfrak{f}:[s_0,L]\to \mathbb{R}:\mathfrak{f}\mathrm{is}\mathrm{continuous}.\right\},$$

which is a Banach space with the norm

$$\parallel \mathfrak{f}\parallel =\underset{s\in [s_0,L]}{sup}\left\{\right|\mathfrak{f}\left(s\right)|:\forall \mathfrak{f}\in \widehat{F}\}.$$

Based on the conclusion of Lemma 3, this is the same position that we should define a nonlinear operator like $\widehat{\mathrm{{\rm Y}}}:\widehat{F}\to \widehat{F}$ according to the structure of the solution of the given generalized $(p,q)$-$\varphi$-difference problem (18), where $\widehat{F}=C([s_0,L],\mathbb{R})$.

In other words, we define

$$\begin{array}{cc}\hfill \left(\widehat{\mathrm{{\rm Y}}}\mathfrak{f}\right)\left(s\right)& :=\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,\mathfrak{f}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),\hfill \\ \hfill & +\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right]\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,\mathfrak{f}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),\hfill \\ \hfill & ={\int }_{s_0}^s\mathfrak{h}(u,\mathfrak{f}\left(u\right)){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda }{1-\lambda }{\int }_{s_0}^L\mathfrak{h}(u,\mathfrak{f}\left(u\right)){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u\hfill \end{array}$$

(23)

for all $\mathfrak{f}\in \widehat{F}$ and $s\in [s_0,L]$, where $\lambda \in {\mathbb{R}}^{+}$ with $\lambda <1$.

The Banach contraction mapping principle helps us now to state and prove our main theorem about the existence of unique solutions for the given generalized $(p,q)$-$\varphi$-difference problem (18).

Theorem 9. 

For each ${\mathfrak{f}}_1,{\mathfrak{f}}_2\in \mathbb{R}$, let the nonlinear mapping $\mathfrak{h}:[s_0,L]\times \mathbb{R}\to \mathbb{R}$ satisfy the Lipschitz inequality

$$|\mathfrak{h}(s,{\mathfrak{f}}_1)-\mathfrak{h}(s,{\mathfrak{f}}_2)|\le M_{\mathfrak{h}}|{\mathfrak{f}}_1-{\mathfrak{f}}_2|$$

(24)

for some constant $M_{\mathfrak{h}}>0$. If $\lambda +M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))<1$, then the generalized $(p,q)$-ϕ-difference problem (18) has a unique solution $\mathfrak{f}\left(s\right)$ for all $s\in [s_0,L]$.

Proof. 

Define a ball by $\mathcal{B}:=\{\mathfrak{f}\in \widehat{F}:\parallel \mathfrak{f}\parallel \le \mathsf{ϵ}\}$ with the positive radius $\mathsf{ϵ}$, which is satisfied by

$$\mathsf{ϵ}>\frac{{\mathfrak{h}}^{*}(\varphi \left(L\right)-\varphi \left(s_0\right))}{1-(\lambda +M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right)))},$$

so that ${\mathfrak{h}}^{*}:=sup\left\{\right|\mathfrak{h}(s,0):s\in [s_0,L]\}$.

Assume that $\mathfrak{f}\in \mathcal{B}$ and $s\in [s_0,L]$ are arbitrary. By the Lipschitz Inequality (24), we have

$$\begin{array}{cc}\hfill \left|\right(\widehat{\mathrm{{\rm Y}}}& \mathfrak{f}\left)\left(s\right)\right|\le \sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right]|\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,\mathfrak{f}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),|\hfill \\ \hfill & +\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right]|\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,\mathfrak{f}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),|\hfill \\ \hfill & ={\int }_{s_0}^s\left|\mathfrak{h}(u,\mathfrak{f}\left(u\right))\right|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda }{1-\lambda }{\int }_{s_0}^L\left|\mathfrak{h}(u,\mathfrak{f}\left(u\right))\right|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le {\int }_{s_0}^s\left(\left|\mathfrak{h}\right(u,\mathfrak{f}\left(u\right))-\mathfrak{h}(s,0\left)\right|+\left|\mathfrak{h}\right(s,0\left)\right|\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & +\frac{\lambda }{1-\lambda }{\int }_{s_0}^L\left(\left|\mathfrak{h}\right(u,\mathfrak{f}\left(u\right))-\mathfrak{h}(s,0\left)\right|+\left|\mathfrak{h}\right(s,0\left)\right|\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le {\int }_{s_0}^s\left(M_{\mathfrak{h}}\left|\mathfrak{f}\left(u\right)\right|+{\mathfrak{h}}^{*}\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda }{1-\lambda }{\int }_{s_0}^L\left(M_{\mathfrak{h}}\left|\mathfrak{f}\left(u\right)\right|+{\mathfrak{h}}^{*}\right){}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le (M_{\mathfrak{h}}\mathsf{ϵ}+{\mathfrak{h}}^{*}){\int }_{s_0}^s{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda (M_{\mathfrak{h}}\mathsf{ϵ}+{\mathfrak{h}}^{*})}{1-\lambda }{\int }_{s_0}^L{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le (M_{\mathfrak{h}}\mathsf{ϵ}+{\mathfrak{h}}^{*})(\varphi \left(L\right)-\varphi \left(s_0\right))+\frac{\lambda (M_{\mathfrak{h}}\mathsf{ϵ}+{\mathfrak{h}}^{*})}{1-\lambda }(\varphi \left(L\right)-\varphi \left(s_0\right)),\hfill \\ \hfill & =\frac{(M_{\mathfrak{h}}\mathsf{ϵ}+{\mathfrak{h}}^{*})}{1-\lambda }(\varphi \left(L\right)-\varphi \left(s_0\right))<\mathsf{ϵ}.\hfill \end{array}$$

Therefore, $\widehat{\mathrm{{\rm Y}}}\left(\mathcal{B}\right)\subseteq \mathcal{B}$. In the following, the operator $\widehat{\mathrm{{\rm Y}}}$ should be a contraction. To prove such a claim, we assume that the arbitrary members ${\mathfrak{f}}_1$ and ${\mathfrak{f}}_2$ are contained in the ball $\mathcal{B}$ with the positive radius $\mathsf{ϵ}$. Then, we may estimate as follows:

$$\begin{array}{cc}\hfill |\left(\widehat{\mathrm{{\rm Y}}}{\mathfrak{f}}_1\right)\left(s\right)& -\left(\widehat{\mathrm{{\rm Y}}}{\mathfrak{f}}_2\right)\left(s\right)|\le \sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(s\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(s\right)\right)\right],\hfill \\ \hfill & \times |\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,{\mathfrak{f}}_1\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),\hfill \\ \hfill & -\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,{\mathfrak{f}}_2\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}s+(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}})s_0\right)\right),|\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\frac{\lambda }{1-\lambda }\sum _{\mathsf{\ell }=0}^{\infty }\left[\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }}\left(L\right)\right)-\varphi \left({}_{s_0}A_{\frac{q}{p}}^{\mathsf{\ell }+1}\left(L\right)\right)\right],\hfill \\ \hfill & \times |\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,{\mathfrak{f}}_1\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}})s_0\right)\right),\hfill \\ \hfill & -\mathfrak{h}\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0,{\mathfrak{f}}_2\left(\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}L+\left(1-\frac{q^{\mathsf{\ell }}}{p^{\mathsf{\ell }+1}}\right)s_0\right)\right),|\hfill \\ \hfill & \le {\int }_{s_0}^s|\mathfrak{h}(u,{\mathfrak{f}}_1\left(u\right))-\mathfrak{h}(u,{\mathfrak{f}}_2\left(u\right))|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda }{1-\lambda }{\int }_{s_0}^L|\mathfrak{h}(u,{\mathfrak{f}}_1\left(u\right))-\mathfrak{h}(u,{\mathfrak{f}}_2\left(u\right))|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le {\int }_{s_0}^s{M}_{\mathfrak{h}}|{\mathfrak{f}}_1\left(u\right)-{\mathfrak{f}}_2\left(u\right)|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda }{1-\lambda }{\int }_{s_0}^L{M}_{\mathfrak{h}}|{\mathfrak{f}}_1\left(u\right)-{\mathfrak{f}}_2\left(u\right)|{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le M_{\mathfrak{h}}\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel {\int }_{s_0}^s{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u+\frac{\lambda M_{\mathfrak{h}}\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel }{1-\lambda }{\int }_{s_0}^L{}_{s_0}\mathrm{d}_{p,q}^{\varphi }u,\hfill \\ \hfill & \le M_{\mathfrak{h}}\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel (\varphi \left(L\right)-\varphi \left(s_0\right))+\frac{\lambda M_{\mathfrak{h}}\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel }{1-\lambda }(\varphi \left(L\right)-\varphi \left(s_0\right)),\hfill \\ \hfill & =\left(\frac{M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))}{1-\lambda }\right)\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel .\hfill \end{array}$$

From the above inequalities, one can find out that

$$\parallel \left(\widehat{\mathrm{{\rm Y}}}{\mathfrak{f}}_1\right)-\left(\widehat{\mathrm{{\rm Y}}}{\mathfrak{f}}_2\right)\parallel \le \left(\frac{M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))}{1-\lambda }\right)\parallel {\mathfrak{f}}_1-{\mathfrak{f}}_2\parallel ,$$

implying the fact that $\widehat{\mathrm{{\rm Y}}}$ is a contraction on Ball $\mathcal{B}$ since $\lambda +M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))<1$. Therefore, in referring to the Banach contraction mapping principle, we find that our defined operator $\widehat{\mathrm{{\rm Y}}}$ has a unique fixed point on $\mathcal{B}$. In other words, the given generalized $(p,q)$-$\varphi$-difference problem (18) contains a unique solution. This completes the proof of the theorem. □

5. One Example

Validation of the theoretical results is one the most important sections of every paper, and, in this example, we tried to complete this validation by considering numerical data.

Example 3. 

The following generalized $(p,q)$-ϕ-difference problem

$$\left\{\begin{array}{c}{}_0{}^s\mathbf{D}_{\frac{1}{2},\frac{1}{4}}^{\varphi }\mathfrak{f}\left(s\right)=\frac{4}{7s+5}\left(\frac{\left|\mathfrak{f}\right(s\left)\right|}{1+\left|\mathfrak{f}\right(s\left)\right|}\right)-\frac{1}{8},s\in [0,1],\hfill \\ \mathfrak{f}\left(0\right)=\frac{1}{20}\mathfrak{f}\left(1\right)\hfill \end{array}\right.$$

(25)

is considered with these assumptions: $p=\frac{1}{2}$, $q=\frac{1}{4}$, $s_0=0$, $L=1$, $\lambda =\frac{1}{20}$, and

$$\mathfrak{h}(s,\mathfrak{f}\left(s\right))=\frac{4}{7s+5}\left(\frac{\left|\mathfrak{f}\right(s\left)\right|}{1+\left|\mathfrak{f}\right(s\left)\right|}\right)-\frac{1}{8},s\in [0,1].$$

We can easily see that the Lipschitz inequality is satisfied for the nonlinear function $\mathfrak{h}$ since

$$\begin{array}{cc}\hfill |\mathfrak{h}(s,{\mathfrak{f}}_1\left(s\right))-\mathfrak{h}(s,{\mathfrak{f}}_2\left(s\right))|& \le \frac{4}{7s+5}\left(|\frac{|{\mathfrak{f}}_1\left(s\right)|}{1+|{\mathfrak{f}}_1\left(s\right)|}-\frac{|{\mathfrak{f}}_2\left(s\right)|}{1+|{\mathfrak{f}}_2\left(s\right)|}|\right),\hfill \\ \hfill & \le \frac{4}{7s+5}|{\mathfrak{f}}_1\left(s\right)-{\mathfrak{f}}_2\left(s\right)|,{\mathfrak{f}}_1,{\mathfrak{f}}_2\in \mathbb{R}.\hfill \end{array}$$

Therefore, for each $s\in [0,1]$, we obtained

$$|\mathfrak{h}(s,{\mathfrak{f}}_1)-\mathfrak{h}(s,{\mathfrak{f}}_2)|\le \frac{4}{5}|{\mathfrak{f}}_1-{\mathfrak{f}}_2|,{\mathfrak{f}}_1,{\mathfrak{f}}_2\in \mathbb{R}.$$

Then, we considered two cases for the strictly increasing function $\varphi :[0,1]\to \mathbb{R}$:

Case 1:  $\varphi \left(s\right)=s^2$: Assuming $M_{\mathfrak{h}}=\frac{4}{5}$, the inequality

$$\lambda +M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))=\frac{1}{20}+\frac{4}{5}(\varphi \left(1\right)-\varphi \left(0\right))=\frac{1}{20}+\frac{4}{5}(1-0)=\frac{17}{20}\simeq 0.85<1$$

is satisfied.

Case 2:  $\varphi \left(s\right)=\frac{exp\left(s\right)}{10}$: Assuming $M_{\mathfrak{h}}=\frac{4}{5}$, the inequality

$$\lambda +M_{\mathfrak{h}}(\varphi \left(L\right)-\varphi \left(s_0\right))=\frac{1}{20}+\frac{4}{5}(\varphi \left(1\right)-\varphi \left(0\right))=\frac{1}{20}+\frac{4}{5}(\frac{e}{10}-\frac{1}{10})\simeq 0.1868<1$$

is satisfied.

In both of the cases for the strictly increasing function ϕ, Theorem 9 guarantees that the generalized $(p,q)$-ϕ-difference problem (25) has a unique solution on $[0,1]$.

6. Conclusions

This research focused on the study of $(p,q)$-calculus with respect to a strictly increasing real-valued function like $\varphi$. In this direction, we obtained help from the shifting operators to define the $(p,q)$-$\varphi$-derivatives and $(p,q)$-$\varphi$-integrals. Next, we proved some of the important properties in the framework of several theorems. More precisely, the performance of the $(p,q)$-$\varphi$-derivatives on the power functions was studied. We showed that the generalized $(p,q)$-$\varphi$-derivatives and $(p,q)$-$\varphi$-integrals had a linear property and we proved some results about the acting generalized $(p,q)$-$\varphi$-derivatives on the quotient functions and product functions. Next, the higher-order $(p,q)$-$\varphi$-derivatives along with the double $(p,q)$-$\varphi$-integrals were defined with some examples. Also, an important theorem was provided about the $(p,q)$-$\varphi$-integration by parts. Moreover, we mentioned some of the special cases obtained by these newly defined generalized $(p,q)$-$\varphi$-operators. Finally, as an application, we considered a generalized $(p,q)$-$\varphi$-difference problem involving the $(p,q)$-$\varphi$-derivative, as well as a related numerical example, to show the existence of unique solutions via the Banach contraction mapping principle. In fact, this example showed that we can simulate other forms of the boundary value problems due to real-world phenomena with the help of these newly defined operators with the least errors. All of the results of this paper are new and extend the existing concepts in q-calculus and $(p,q)$-calculus. In further studies, we will try to introduce some numerical algorithms based on these newly defined $(p,q)$-$\varphi$-operators for the purpose of finding approximate solutions.