Skip to main content

Existence results of nonlocal Robin boundary value problems for fractional \((p,q)\)-integrodifference equations

Abstract

The existence results of a fractional \((p,q)\)-integrodifference equation with nonlocal Robin boundary condition are investigated by using Banach’s and Schauder’s fixed point theorems. Moreover, we study some properties of \((p,q)\)-integral that will be used as a tool for our calculations.

Introduction

Along with the evolution of the theory and application of classical calculus, quantum calculus (calculus without limit) has also received more intense attention in the last three decades. In this article, we study the development of q-calculus which is one type of quantum calculus. The q-calculus was first introduced by Jackson [1, 2] in 1910. In recent years, the extension of this topic has been studied by many researchers and has many new results in [39] and their references. The knowledge of q-calculus was used in physical problems, see [1027] and the references cited therein.

Later, the study of quantum calculus based on two-parameter \((p, q)\)-integer was presented. The \((p,q)\)-calculus was presented by Chakrabarti and Jagannathan [27]. The extension of studies of \((p,q)\)-calculus was given in [2839]. In addition, it is used in many branches such as physical sciences, hypergeometric series, Lie group, special functions, approximation theory, Bézier curves and surfaces, etc. [4047].

Then, the study of fractional quantum calculus was initiated [4850]. Agarwal [48] and Al-Salam [49] studied fractional q-calculus, whilst Díaz and Osler [50] proposed fractional difference calculus. In 2017, Brikshavana and Sitthiwirattham [51] introduced fractional Hahn difference calculus. Recently, Patanarapeelert and Sitthiwirattham [52] studied fractional symmetric Hahn difference calculus. Presently, Soontharanon and Sitthiwirattham [53] introduced the fractional \((p,q)\)-difference operators and their properties.

There are some recent papers studying the boundary value problem for \((p, q)\)-difference equations [5456]. However, the boundary value problem for fractional \((p, q)\)-difference equations has not been studied since fractional \((p, q)\)-operators have been defined lately. These motivate the authors for this research. This article investigates the existence results of a fractional \((p,q)\)-integrodifference equation with nonlocal Robin boundary value conditions of the form

$$\begin{aligned}& \begin{gathered} D^{\alpha}_{p,q} u(t)=F \bigl[ t,u(t),\varPsi ^{\gamma }_{p,q}u(t),D^{\nu }_{p,q}u(t) \bigr],\quad t\in I^{T}_{p,q}, \\ \lambda _{1}u(\eta )+\lambda _{2}D_{p,q}^{\beta }u( \eta )=\phi _{1}(u), \quad \eta \in I^{T}_{p,q} - \biggl\lbrace 0,\frac{T}{p} \biggr\rbrace , \\ \mu _{1}u \biggl(\frac{T}{p} \biggr) +\mu _{2}D_{p,q}^{\beta }u \biggl( \frac{T}{p} \biggr) =\phi _{2}(u), \end{gathered} \end{aligned}$$
(1.1)

where \(I^{T}_{p,q}:= \lbrace \frac{q^{k}}{p^{k+1}}T:k \in {\mathbb{N}}_{0} \rbrace \cup \{0\}\); \(0< q< p\leq 1\)\(\alpha \in (1,2]\), \(\beta ,\gamma ,\nu \in (0,1]\), \(\lambda _{1},\lambda _{2},\mu _{1},\mu _{2} \in {{\mathbb{R}}^{+}}\); \(F \in C (I_{p,q}^{T}\times {\mathbb{R}}\times {\mathbb{R}}\times { \mathbb{R}},{\mathbb{R}} )\) is a given function; \(\phi _{1},\phi _{2} : C (I_{p,q}^{T},{\mathbb{R}} ) \rightarrow {\mathbb{R}}\) are given functionals; and for \(\varphi \in C (I_{p,q}^{T}\times I_{p,q}^{T},[0,\infty ) )\), we define an operator of the \((p,q)\)-integral of the product of functions φ and u as

$$\begin{aligned} \varPsi ^{\gamma }_{p,q} u(t):= \bigl({\mathcal{I}}_{p,q}^{\gamma } \varphi u \bigr) (t)=\frac{1}{p^{{\gamma \choose 2}}\varGamma _{p,q}(\gamma )} \int _{0}^{t} ( t-qs) ^{\underline{\gamma -1}}_{p,q} \varphi (t,s) u \biggl( \frac{s}{p^{\gamma -1}} \biggr) \, d_{p,q}s. \end{aligned}$$

We aim to prove the existence and uniqueness of a solution for this problem by using Banach’s fixed point theorem, and the existence of at least one solution by using Schauder’s fixed point theorem. In addition, we provide an example to illustrate our results.

Preliminaries

In this section, we recall some basic definitions, notations, and lemmas. Letting \(0< q< p\leq 1\), we define the notations

$$\begin{aligned}& {[k]}_{q}:= \textstyle\begin{cases} \frac{1-q^{k}}{1-q}, & k \in \mathbb{N} \\ 1, & k=0, \end{cases}\displaystyle \\& {[k]}_{p,q}:= \textstyle\begin{cases} \frac{p^{k}-q^{k}}{p-q}=p^{k-1}[k]_{\frac{q}{p}}, & k \in \mathbb{N} \\ 1, & k=0, \end{cases}\displaystyle \\& {[k]}_{p,q}!:= \textstyle\begin{cases} {[k]}_{p,q}{[k-1]}_{p,q}\cdot \cdot \cdot {[1]}_{p,q}= \prod_{i=1}^{k}\frac{p^{i}-q^{i}}{p-q}, & k \in \mathbb{N} \\ 1, & k=0. \end{cases}\displaystyle \end{aligned}$$

The \((p,q)\)-forward jump and the \((p,q)\)-backward jump operators are defined as

$$\sigma ^{k}_{p,q}(t):= \biggl(\frac{q}{p} \biggr)^{k}t \quad \mbox{and}\quad \rho ^{k}_{p,q}(t):= \biggl(\frac{p}{q} \biggr)^{k}t\quad \mbox{for } k\in { \mathbb{N}},\mbox{ respectively}. $$

The q-analogue of the power function \({(a-b)}_{q}^{\underline{n}}\) with \(n\in \mathbb{N}_{0}:=\{0,1,2,\ldots\}\) is given by

$$ (a-b)_{q}^{\underline{0}}:=1,\qquad (a-b)_{q}^{\underline{n}}:= \prod_{i=0}^{n-1} \bigl(a-bq^{i} \bigr),\quad a,b\in \mathbb{R}. $$

The \((p,q)\)-analogue of the power function \((a-b)_{p,q}^{\underline{n}}\) with \(n\in \mathbb{N}_{0}\) is given by

$$ (a-b)_{p,q}^{\underline{0}}:=1,\qquad (a-b)_{p,q}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl(ap^{k}-bq^{k} \bigr),\quad a,b\in \mathbb{R}. $$

For \(\alpha \in \mathbb{R}\), we define a general form:

$$\begin{aligned}& (a-b)_{q}^{\underline{\alpha}}= a^{\alpha}\prod _{i=0}^{ \infty } \frac{1- (\frac{b}{a} )q^{i}}{1- (\frac{b}{a} )q^{\alpha +i}},\quad a\neq 0. \\& {(a-b)}_{p,q}^{\underline{\alpha}}= p^{{\alpha \choose 2}}(a-b)_{ \frac{q}{p}}^{\underline{\alpha}} = a^{\alpha}\prod_{i=0}^{ \infty } \frac{1}{p^{\alpha}} \biggl[ \frac{1-\frac{b}{a} ( \frac{q}{p} ) ^{i} }{1-\frac{b}{a} ( \frac{q}{p} ) ^{i+\alpha }} \biggr],\quad a\neq 0. \end{aligned}$$

Note that \(a_{q}^{\underline{\alpha}} = a^{\alpha}\), \({a}_{p,q}^{ \underline{\alpha}} = ( \frac{a}{p} ) ^{\alpha}\) and \((0)_{q}^{\underline{\alpha}}={(0)}_{p,q}^{\underline{\alpha}}=0\) for \(\alpha >0\).

The \((p,q)\)-gamma and \((p,q)\)-beta functions are defined by

$$\begin{aligned}& {\varGamma }_{p,q}(x):= \textstyle\begin{cases} \frac{(p-q)_{p,q}^{\underline{x-1}}}{(p-q)^{x-1}} = \frac{ (1-\frac{q}{p} )_{p,q}^{\underline{x-1}}}{ (1-\frac{q}{p} )^{x-1}}, & x\in \mathbb{R}\setminus \{0,-1,-2,\ldots\} \\ {[x-1]}_{p,q}!, & x\in \mathbb{N} \end{cases}\displaystyle \\& {B}_{p,q}(x,y):= \int _{0}^{1} t^{x-1}(1-qt)_{p,q}^{\underline{y-1}} \,d_{p,q}t= p^{\frac{1}{2}(y-1)(2x+y-2)} \frac{{\varGamma }_{p,q}(x){\varGamma }_{q}{p,q}(y)}{{\varGamma }_{p,q}(x+y)}, \end{aligned}$$

respectively.

Lemma 2.1

([53])

For\(\alpha ,\beta ,\gamma ,\lambda \in {\mathbb{R}}\),

$$\begin{aligned} \mathrm{(a)} & (\gamma \beta -\gamma \lambda )_{p,q}^{\underline{\alpha}}= \gamma ^{\alpha}(\beta -\lambda )_{p,q}^{\underline{\alpha}}, \\ \mathrm{(b)} & (\beta -\gamma )_{p,q}^{\underline{\alpha +\gamma }}= \frac{1}{p^{\alpha \gamma }}(\beta -\gamma )_{p,q}^{\underline{\alpha}} \bigl(p^{\alpha}\beta -q^{\alpha}\lambda \bigr)_{p,q}^{\underline{\gamma }}, \\ \mathrm{(c)} & (t-s)_{p,q}^{\underline{\alpha}}=0, \quad \alpha \notin {\mathbb{N}}_{0}, t \geq s,\textit{ and }t,s \in I_{p,q}^{T} . \end{aligned}$$

Lemma 2.2

([53])

For\(m,n\in {\mathbb{N}}_{0}\), \(\alpha \in \mathbb{R}\), and\(0< q< p\leq 1\),

$$\begin{aligned} \mathrm{(a)} & { \bigl(t-\sigma _{p,q}^{n}(t) \bigr)}_{p,q}^{\underline{\alpha}}=t^{\alpha} \biggl( 1- \biggl( \frac{q}{p} \biggr) ^{n} \biggr) _{p,q}^{ \underline{\alpha}}, \\ \mathrm{(b)} & { \bigl(\sigma _{p,q}^{m}(t)- \sigma _{p,q}^{n}(t) \bigr)}_{p,q}^{ \underline{\alpha}}= \biggl( \frac{q}{p} \biggr) ^{m\alpha }t^{\alpha} \biggl( 1- \biggl( \frac{q}{p} \biggr) ^{n-m} \biggr)_{p,q}^{ \underline{\alpha}}. \end{aligned}$$

Definition 2.1

For \(0< q< p\leq 1\) and \(f:[0,T]\rightarrow {\mathbb{R}}\), we define the \((p,q)\)-difference of f as

$$\begin{aligned} D_{p,q}f(t) :=& \textstyle\begin{cases} \frac{f(pt )-f(qt)}{(p-q)(t)}, & \mbox{for }t\neq 0 \\ f'(0), & \text{for }t=0 \end{cases}\displaystyle \end{aligned}$$

provided that f is differentiable at 0. f is called \((p,q)\)-differentiable on \(I_{p,q}^{T}\) if \(D_{p,q}f(t)\) exists for all \(t \in I_{p,q}^{T}\).

Lemma 2.3

([31])

Letf, gbe\((p,q)\)-differentiable on\(I_{p,q}^{T}\). The properties of\((p,q)\)-difference operator are as follows:

$$\begin{aligned} \mathrm{(a)} & D_{p,q}\bigl[f(t)+g(t) \bigr]=D_{p,q}f(t)+D_{p,q}g(t), \\ \mathrm{(b)} & D_{p,q}\bigl[\alpha f(t)\bigr]=\alpha D_{p,q}f(t)\quad \textit{for } \alpha \in {\mathbb{R}}, \\ \mathrm{(c)} & D_{p,q}\bigl[f(t)g(t)\bigr]=f(pt)D_{p,q}g(t)+g(qt)D_{p,q}f(t)=g(pt)D_{p,q}f(t)+f(qt)D_{p,q}g(t), \\ \mathrm{(d)} & D_{p,q} \biggl[ \frac{f(t)}{g(t)} \biggr] = \frac{g(qt)D_{p,q}f(t)-f(qt)D_{p,q}g(t)}{g(pt)g(qt)} = \frac{g(pt)D_{p,q}f(t)-f(pt)D_{p,q}g(t)}{g(pt)g(qt)}\\ &\quad \textit{for } g(pt)g(qt)\neq 0. \end{aligned}$$

Lemma 2.4

([53])

Let\(t\in I_{p,q}^{T}\), \(0< q< p\leq 1\), \(\alpha \geq 1\), and\({a \in \mathbb{R}}\). Then

$$\begin{aligned} \mathrm{(a)} & D_{p,q}(t-a)_{p,q}^{\underline{\alpha}}=[ \alpha ]_{p,q} ( pt-a ) _{p,q}^{\underline{\alpha -1}}, \\ \mathrm{(b)} & D_{p,q}(a-t)_{p,q}^{\underline{\alpha}}=-[ \alpha ]_{p,q} ( a-qt ) _{p,q}^{\underline{\alpha -1}}. \end{aligned}$$

Definition 2.2

Let I be any closed interval of \(\mathbb{R}\) containing a, b, and 0. Assuming that \(f:I\rightarrow \mathbb{R}\) is a given function, we define \((p,q)\)-integral of f from a to b by

$$ \int _{a}^{b} f(t)\,d_{p,q}t:= \int _{0}^{b} f(t)\,d_{p,q}t- \int _{0}^{a} f(t)\,d_{p,q}t, $$

where

$$\begin{aligned} {\mathcal{I}}_{p,q}f(x)= \int _{0}^{x} f(t)\,d_{p,q}t= (p-q)x \sum_{k=0}^{\infty } \frac{q^{k}}{p^{k+1}} f \biggl( \frac{q^{k}}{p^{k+1}}x \biggr),\quad x\in I, \end{aligned}$$

provided that the series converges at \(x=a\) and \(x=b\). f is called \((p,q)\)-integrable on \([a,b]\) if it is \((p,q)\)-integrable on \([a,b]\) for all \(a,b\in I\).

Next, we define an operator \({{\mathcal{I}}}_{p,q}^{N} \) as

$$ {{\mathcal{I}}}_{p,q}^{0}f(x)=f(x)\quad \text{and}\quad {{\mathcal{I}}}_{p,q}^{N} f(x)={{ \mathcal{I}}}_{p,q} {{\mathcal{I}}}_{p,q}^{N-1} f(x),\quad N\in { \mathbb{N}}. $$

The relations between \((p,q)\)-difference and \((p,q)\)-integral operators are given by

$$ {D}_{p,q}{{\mathcal{I}}}_{p,q}f(x)=f(x)\quad \text{and}\quad {{\mathcal{I}}}_{p,q}{D}_{p,q} f(x)=f(x)-f(0). $$

Lemma 2.5

([31])

Let\(0< q< p\leq 1\), \(a,b \in I_{p,q}^{T}\), andf, gbe\((p,q)\)-integrable on\(I_{p,q}^{T}\). Then the following formulas hold:

$$\begin{aligned} \mathrm{(a)} & \int _{a}^{a} f(t)\,d_{p,q}t=0, \\ \mathrm{(b)} & \int _{a}^{b} \alpha f(t)\,d_{p,q}t= \alpha \int _{a}^{b} f(t)\,d_{p,q}t,\quad \alpha \in {\mathbb{R}}, \\ \mathrm{(c)} & \int _{a}^{b} f(t)\,d_{p,q}t=- \int _{b}^{a} f(t)\,d_{p,q}t, \\ \mathrm{(d)} & \int _{a}^{b} f(t)\,d_{p,q}t= \int _{c}^{b} f(t)\,d_{p,q}t+ \int _{a}^{c} f(t)\,d_{p,q}t,\quad c\in I_{p,q}^{T}, a< c< b, \\ \mathrm{(e)} & \int _{a}^{b} \bigl[ f(t)+g(t) \bigr] \,d_{p,q}t= \int _{a}^{b} f(t)\,d_{p,q}t+ \int _{a}^{b} g(t)\,d_{p,q}t, \\ \mathrm{(f)} & \int _{a}^{b} \bigl[ f(pt)D_{p,q}g(t) \bigr] \,d_{p,q}t= \bigl[ f(t)g(t) \bigr] _{a}^{b}- \int _{a}^{b} \bigl[g(qt)D_{p,q}f(t) \bigr] \,d_{p,q}t. \end{aligned}$$

Lemma 2.6

([31], Fundamental theorem of \((p,q)\)-calculus)

Letting\(f:I\rightarrow \mathbb{R}\)be continuous at 0 and

$$ F(x):= \int _{0}^{x} f(t)\,d_{p,q}t,\quad x\in I, $$

thenFis continuous at 0 and\(D_{p,q}F(x)\)exists for every\(x\in I\)where

$$ D_{p,q}F(x)=f(x). $$

Conversely,

$$ \int _{a}^{b} D_{p,q}f(t) \,d_{p,q}t=f(b)-f(a) \quad \textit{for all }a,b\in I. $$

Lemma 2.7

([53], Leibniz formula of \((p,q)\)-calculus)

Letting\(f:I_{p,q}^{T}\times I_{p,q}^{T}\rightarrow \mathbb{R}\),

$$ D_{p,q} \biggl[ \int _{0}^{t} f(t,s) \,d_{p,q}s \biggr] = \int _{0}^{qt} {_{t}}D_{p,q}f(t,s) \,d_{p,q}s+f ( pt,t ), $$

where\({_{t}}D_{p,q}\)is\((p,q)\)-difference with respect tot.

Next we introduce fractional \((p,q)\)-integral and fractional \((p,q)\)-difference of Riemann–Liouville type as follows.

Definition 2.3

For \(\alpha >0\), \(0< q< p\leq 1\), and f defined on \(I^{T}_{p,q}\), the fractional \((p,q)\)-integral is defined by

$$\begin{aligned} {\mathcal{I}}_{p,q}^{\alpha}f(t) :=& \frac{1}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{t} (t-qs)_{p,q}^{\underline{\alpha -1}}f \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ =&\frac{ (p-q)t }{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \sum_{k=0}^{\infty } \frac{q^{k}}{p^{k+1}} \biggl( t- \biggl(\frac{q}{p} \biggr)^{k+1}t \biggr) _{p,q}^{ \underline{\alpha -1}} f \biggl( \frac{q^{k}}{p^{k+\alpha }}t \biggr), \end{aligned}$$

and \(({\mathcal{I}}^{0}_{p,q} f)(t) = f(t)\).

Definition 2.4

For \(\alpha >0\), \(0< q< p\leq 1\), and f defined on \(I^{T}_{p,q}\), the fractional \((p,q)\)-difference operator of Riemann–Liouville type of order α is defined by

$$\begin{aligned} D_{p,q}^{\alpha}f(t) :=&D_{p,q}^{N} {\mathcal{I}}_{p,q}^{N-\alpha } f(t) \\ =&\frac{1}{p^{{-\alpha \choose 2}}\varGamma _{p,q}(-\alpha )} \int _{0}^{t} (t-qs)_{p,q}^{\underline{-\alpha -1}}f \biggl( \frac{s}{p^{-\alpha -1}} \biggr) \,d_{p,q}s, \end{aligned}$$

and \(D^{0}_{p,q}f(t) = f(t)\), where \(N-1<\alpha < N\), \(N\in {\mathbb{N}}\).

Lemma 2.8

([53])

Letting\(\alpha \in (N-1,N)\), \(N\in {\mathbb{N}}\), \(0< q< p\leq 1\), and\(f:I_{p,q}^{T}\rightarrow {\mathbb{R}}\),

$$\begin{aligned} \mathcal{I}_{p,q}^{\alpha}D_{p,q}^{\alpha}f(t) =f(t)+C_{1}t^{\alpha -1}+C_{2}t^{ \alpha -2}+ \cdots +C_{N}t^{\alpha -N} \end{aligned}$$

for some\(C_{i}\in {\mathbb{R}}\), \(i=1,2,\ldots,N\).

Lemma 2.9

([53])

Letting\(0< q< p\leq 1\)and\(f:I_{p,q}^{T}\rightarrow \mathbb{R}\)be continuous at 0,

$$\begin{aligned} \int _{0}^{x} \int _{0}^{s} f(\tau ) \,d_{p,q} \tau \,d_{p,q}s = \int _{0}^{\frac{x}{p}} \int _{pq\tau }^{x} f(\tau ) \,d_{p,q}s \,d_{p,q} \tau . \end{aligned}$$

Lemma 2.10

([53])

Let\(\alpha ,\beta >0\), \(0< q< p\leq 1\). Then

$$\begin{aligned} \mathrm{(a)} & \int _{0}^{t} ( t-qs)_{p,q}^{\underline{\alpha -1}} s^{\beta } \,d_{p,q}s = t^{\alpha +\beta }B_{p,q}( \beta +1,\alpha ), \\ \mathrm{(b)} & \int _{0}^{t} \int _{0}^{x}(t-qx)_{p,q}^{\underline{\alpha -1}} (x-qs)_{p,q}^{ \underline{\beta -1}} \,d_{p,q}s \,d_{p,q}x = \frac{ B_{p,q} (\beta +1,\alpha ) }{[\beta ]_{p,q}}t^{\alpha +\beta }. \end{aligned}$$

Lemma 2.11

Let\(\alpha ,\beta >0\), \(0< q< p\leq 1\), and\(n \in \mathbb{Z}\). Then

$$\begin{aligned} \mathrm{(a)} & \int _{0}^{t} ( t-qs)_{p,q}^{\underline{\alpha -1}} \,d_{p,q}s = p^{{\alpha \choose 2}} \frac{ \varGamma _{p,q}(\alpha ) }{\varGamma _{p,q}(\alpha +1)} t^{\alpha}, \\ \mathrm{(b)} & \int _{0}^{t} \int _{0}^{\frac{x}{p^{-\beta -1}}} (t-qx)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr) _{p,q}^{ \underline{\alpha -1}} \,d_{p,q}s \,d_{p,q}x = p^{ {\alpha \choose 2}+{- \beta \choose 2} } \frac{ \varGamma _{p,q}(\alpha ) }{\varGamma _{p,q}(\alpha +1)} t^{ \alpha +\beta }, \\ \mathrm{(c)} & \int _{0}^{t} (t-qs)_{p,q}^{\underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{\alpha -n} \,d_{p,q}s = p^{ {- \beta \choose 2} } \frac{ \varGamma _{p,q}(\alpha -n+1)\varGamma _{p,q}(-\beta )}{\varGamma _{p,q}(\alpha -\beta -n+1)} t^{\alpha -\beta -n}. \end{aligned}$$

Proof

From Lemma 2.10(a) and the definition of \((p,q)\)-beta function, we have

$$\begin{aligned}& \int _{0}^{t} ( t-qs)_{p,q}^{\underline{\alpha -1}} \,d_{p,q}s=t^{ \alpha } B_{p,q}(1,\alpha ) = p^{{\alpha \choose 2}} \frac{ \varGamma _{p,q}(\alpha ) }{\varGamma _{p,q}(\alpha +1)} t^{\alpha}, \\& \int _{0}^{t} \int _{0}^{\frac{x}{p^{-\beta -1}}} (t-qx)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr) _{p,q}^{ \underline{\alpha -1}} \,d_{p,q}s \,d_{p,q}x \\& \quad =\frac{B_{p,q}(1,\alpha ) }{p^{-\alpha (\beta +1)}} \int _{0}^{t} (t-qx)_{p,q}^{ \underline{-\beta -1}} x^{\alpha} \,d_{p,q}x \\& \quad =\frac{B_{p,q}(1,\alpha ) }{p^{-\alpha (\beta +1)}} B_{p,q}(\alpha +1,- \beta ) t^{\alpha -\beta } \\& \quad = p^{ {\alpha \choose 2}+{-\beta \choose 2} } \frac{ \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) }{\varGamma _{p,q}(\alpha -\beta +1)} t^{\alpha -\beta }. \end{aligned}$$

For \(n \in \mathbb{Z}\), we have

$$\begin{aligned} \int _{0}^{t} (t-qs)_{p,q}^{\underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{\alpha -n} \,d_{p,q}s =& \frac{B_{p,q}(\alpha ,-\beta ) }{p^{(\alpha -n)(-\beta -1)}} t^{\alpha -\beta -n+1} \\ =&p^{ {-\beta \choose 2} } \frac{ \varGamma _{p,q}(\alpha -n+1)\varGamma _{p,q}(-\beta )}{\varGamma _{p,q}(\alpha -\beta -n+1)} t^{\alpha -\beta -n}. \end{aligned}$$

The proof is complete. □

We next provide a lemma showing a result of the linear variant of problem (1.1).

Lemma 2.12

Let\(\varOmega \neq 0\), \(\alpha \in (1,2]\), \(\beta \in (0,1]\), \(0< q< p\leq 1\), \(\lambda _{1},\lambda _{2},\mu _{1},\mu _{2} \in {{\mathbb{R}}^{+}}\), \(h \in C (I_{p,q}^{T},\mathbb{R} )\)is a given function; \(\phi _{1},\phi _{2} : C (I_{p,q}^{T},\mathbb{R} )\rightarrow \mathbb{R}\)are given functionals. The linear variant problem of (1.1)

$$\begin{aligned}& D^{\alpha}_{p,q} u(t)=h(t),\quad t\in I^{T}_{p,q}, \\& \lambda _{1}u(\eta )+\lambda _{2}D_{p,q}^{\beta }u( \eta )=\phi _{1}(u), \quad \eta \in I^{T}_{p,q} - \biggl\lbrace \omega _{0},\frac{T}{p} \biggr\rbrace , \\& \mu _{1}u \biggl(\frac{T}{p} \biggr) +\mu _{2}D_{p,q}^{\beta }u \biggl( \frac{T}{p} \biggr) =\phi _{2}(u) \end{aligned}$$
(2.1)

has the unique solution

$$\begin{aligned} u(t) = {} &\frac{1}{ p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha ) } \int _{0}^{t} (t-qs )_{p,q}^{\underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &{}-\frac{t^{\alpha -1}}{\varOmega } \bigl\{ \mathbf{B}_{T}{\varPhi _{\eta }[ \phi _{1},h]} -{\mathbf{B}}_{\eta }{ \varPhi _{T}[\phi _{2},h]} \bigr\} \\ &{}+\frac{t^{\alpha -2}}{\varOmega } \bigl\{ \mathbf{A}_{T}{\varPhi _{\eta }[ \phi _{1},h]} -{\mathbf{A}}_{\eta }{ \varPhi _{T}[\phi _{2},h]} \bigr\} , \end{aligned}$$
(2.2)

where the functionals\({\varPhi _{\eta }[\phi _{1},h]}\), \({\varPhi _{T}[\phi _{2},h]}\)are defined by

$$\begin{aligned} {\varPhi _{\eta }[\phi _{1},h]}:= {}&\phi _{1}(u)- \frac{\lambda _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \eta } (\eta -qs )_{p,q}^{\underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &{}- \frac{\lambda _{2}}{p^{ {\alpha \choose 2} +{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) } \\ &{}\times \int _{0}^{\eta } \int _{0}^{\frac{x}{p^{-\beta -1}}} (\eta -qx)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr)_{p,q}^{ \underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \,d_{p,q}x, \end{aligned}$$
(2.3)
$$\begin{aligned} {\varPhi _{T}[\phi _{2},h]}:= {}&\phi _{2}(u)- \frac{\mu _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &{}- \frac{\mu _{2}}{p^{ {\alpha \choose 2} +{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) } \\ &{} \times \int _{0}^{\frac{T}{p}} \int _{0}^{\frac{x}{p^{-\beta -1}}} \biggl( \frac{T}{p}-qx \biggr) _{p,q}^{\underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr)_{p,q}^{\underline{\alpha -1}} \\ &{} \times h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \,d_{p,q}x, \end{aligned}$$
(2.4)

and the constants\(\mathbf{A}_{\eta }\), \(\mathbf{A}_{T}\), \(\mathbf{B}_{\eta }\), \(\mathbf{B}_{T}\), andΩare defined by

$$\begin{aligned}& \mathbf{A}_{\eta }:=\lambda _{1}\eta ^{\alpha -1}+ \frac{\lambda _{2}}{p^{{-\beta \choose 2}} \varGamma _{p,q}(-\beta )} \int _{0}^{\eta } (\eta -qs )_{p,q}^{\underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{\alpha -1} \,d_{p,q}s \\& \hphantom{\mathbf{A}_{\eta }}=\eta ^{\alpha -1} \biggl( \lambda _{1}+ \frac{\lambda _{2}\varGamma _{p,q}(\alpha )}{\varGamma _{p,q}(\alpha -\beta )} \eta ^{-\beta } \biggr), \end{aligned}$$
(2.5)
$$\begin{aligned}& \mathbf{A}_{T}:= \mu _{1} \biggl( \frac{T}{p} \biggr) ^{\alpha -1}+ \frac{\mu _{2}}{p^{{-\beta \choose 2}} \varGamma _{p,q}(-\beta )} \int _{0}^{ \frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{ \alpha -1} \,d_{p,q}s \\& \hphantom{\mathbf{A}_{T}}= \biggl( \frac{T}{p} \biggr) ^{\alpha -1} \biggl( \mu _{1}+ \frac{\mu _{2}\varGamma _{p,q}(\alpha )}{\varGamma _{p,q}(\alpha -\beta )} \biggl( \frac{T}{p} \biggr) ^{-\beta } \biggr), \end{aligned}$$
(2.6)
$$\begin{aligned}& \mathbf{B}_{\eta }:= \lambda _{1}\eta ^{\alpha -2}+ \frac{\lambda _{2}}{p^{{-\beta \choose 2}} \varGamma _{p,q}(-\beta )} \int _{0}^{\eta } (\eta -qs )_{p,q}^{\underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{\alpha -2} \,d_{p,q}s \\& \hphantom{\mathbf{B}_{\eta }}=\eta ^{\alpha -2} \biggl( \lambda _{1}+ \frac{\lambda _{2}\varGamma _{p,q}(\alpha -1)}{\varGamma _{p,q}(\alpha -\beta -1)} \eta ^{-\beta } \biggr) , \end{aligned}$$
(2.7)
$$\begin{aligned}& \mathbf{B}_{T}:= \mu _{1} \biggl( \frac{T}{p} \biggr) ^{\alpha -2}+ \frac{\mu _{2}}{p^{{-\beta \choose 2}} \varGamma _{p,q}(-\beta )} \int _{0}^{ \frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{s}{p^{-\beta -1}} \biggr) ^{ \alpha -2} \,d_{p,q}s \\& \hphantom{\mathbf{B}_{T}}= \biggl( \frac{T}{p} \biggr) ^{\alpha -2} \biggl( \mu _{1}+ \frac{\mu _{2}\varGamma _{p,q}(\alpha -1)}{\varGamma _{p,q}(\alpha -\beta -1)} \biggl( \frac{T}{p} \biggr) ^{-\beta } \biggr) , \end{aligned}$$
(2.8)
$$\begin{aligned}& \varOmega := \mathbf{A}_{T}\mathbf{B}_{\eta }- \mathbf{A}_{\eta } \mathbf{B}_{T}. \end{aligned}$$
(2.9)

Proof

To obtain the solution, we first take a fractional \((p,q)\)-integral of order α for (2.1). Then we have

$$\begin{aligned} u(t)&=C_{1}t^{\alpha -1}+C_{2}t^{\alpha -2}+{ \mathcal{I}}^{\alpha}_{p,q}h(t) \\ &=C_{1}t^{\alpha -1}+C_{2}t^{\alpha -2}+ \frac{1}{p^{{ \alpha \choose 2 }}\varGamma _{p,q}(\alpha )} \int _{0}^{t}(t-qs)_{p,q}^{ \underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s. \end{aligned}$$
(2.10)

Next, we take fractional p, q-difference of order β for (2.10) to get

$$\begin{aligned} D_{p,q}^{\beta }u(t)={} &\frac{1}{p^{{-\beta \choose 2}}\varGamma _{p,q}(-\beta )} (t-qs)_{p,q}^{\underline{\alpha -1}} \biggl[C_{1} \biggl( \frac{s}{p^{\alpha -1}} \biggr)^{\alpha -1}+C_{2} \biggl( \frac{s}{p^{\alpha -1}} \biggr)^{\alpha -2} \biggr] \,d_{p,q}s \\ &{}+ \frac{1}{p^{ {\alpha \choose 2} +{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) } \\ & {}\times \int _{0}^{t} \int _{0}^{\frac{x}{p^{-\beta -1}}} (t-qx)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr)_{p,q}^{ \underline{\alpha -1}} h \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \,d_{p,q}x. \end{aligned}$$
(2.11)

Substituting \(t=\eta \) into (2.10) and (2.11) and employing the first condition of (2.1), we have

$$\begin{aligned} {\mathbf{A}}_{\eta }C_{1}+{ \mathbf{B}}_{\eta }C_{2} = \varPhi _{\eta }[ \phi _{1},h]. \end{aligned}$$
(2.12)

Taking \(t=T\) into (2.10) and (2.11) and employing the second condition of (2.1), we have

$$\begin{aligned} {\mathbf{A}}_{T }C_{1}+{ \mathbf{B}}_{T }C_{2} = \varPhi _{T}[\phi _{2},h]. \end{aligned}$$
(2.13)

Solving \(\text{(2.12)}--\text{(2.13)}\), we find that

$$\begin{aligned} C_{1}= \frac{{\mathbf{B}}_{\eta } {\varPhi _{T} } - {\mathbf{B}}_{T } {\varPhi }_{\eta } }{\varOmega } \quad \text{and}\quad C_{2}= \frac{ {\mathbf{A}}_{T } {\varPhi }_{\eta } - {\mathbf{A}}_{\eta } {\varPhi _{T} } }{\varOmega }, \end{aligned}$$

where \({\varPhi _{\eta }[\phi _{1},h]}\), \({\varPhi _{T}[\phi _{2},h]}\), \(\mathbf{A}_{ \eta }\), \(\mathbf{A}_{T}\), \(\mathbf{B}_{\eta }\), \(\mathbf{B}_{T}\), Ω are defined by (2.3)–(2.9), respectively.

Substituting the constants \(C_{1}\), \(C_{2}\) into (2.10), we obtain (2.2). This completes the proof. □

Existence and uniqueness result

In this section, we use Banach’s fixed point theorem to prove the existence and uniqueness result for problem (1.1). Let \({\mathcal{C}}=C (I_{p,q}^{T}, {\mathbb{R}} )\) be a Banach space of all function u with the norm defined by

$$ \Vert u \Vert _{\mathcal{C}}=\max_{t\in I_{p,q}^{T}} \bigl\{ \bigl\vert u(t) \bigr\vert , \bigl\vert D_{p,q}^{\nu }u(t) \bigr\vert \bigr\} , $$

where \(\alpha \in (1,2]\), \(\beta ,\gamma ,\nu \in (0,1]\), \(0< q< p\leq 1\), \(\lambda _{1},\lambda _{2},\mu _{1},\mu _{2} \in {{\mathbb{R}}^{+}}\). Define an operator \({\mathcal{F}}:{\mathcal{C}}\rightarrow {\mathcal{C}}\) by

$$\begin{aligned} ({\mathcal{F}}u) (t) := {}&\frac{1}{ p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha ) } \int _{0}^{t} (t-qs )_{p,q}^{\underline{\alpha -1}} \\ &{}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \\ &{}-\frac{t^{\alpha -1}}{\varOmega } \bigl\{ \mathbf{B}_{T}{\varPhi _{\eta }^{*}[ \phi _{1},F_{u}]} -{\mathbf{B}}_{\eta }{\varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\} \\ &{}+\frac{t^{\alpha -2}}{\varOmega } \bigl\{ \mathbf{A}_{T}{\varPhi _{\eta }^{*}[ \phi _{1},F_{u}]} -{\mathbf{A}}_{\eta }{\varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\} , \end{aligned}$$
(3.1)

where the functionals \({\varPhi _{\eta }^{*}[\phi _{1},F_{u}]}\), \({\varPhi _{T}^{*}[\phi _{2},F_{u}]}\) are defined by

$$\begin{aligned} &\begin{aligned}[b] {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} :={}&\phi _{1}(u)- \frac{\lambda _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \eta } (\eta -qs )_{p,q}^{\underline{\alpha -1}} \\ &{}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \\ &{}- \frac{\lambda _{2}}{p^{ {\alpha \choose 2} +{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) } \\ &{}\times \int _{0}^{\eta } \int _{0}^{\frac{x}{p^{-\beta -1}}} (\eta -qx)_{p,q}^{ \underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr)_{p,q}^{ \underline{\alpha -1}} \\ &{}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \,d_{p,q}x, \end{aligned} \end{aligned}$$
(3.2)
$$\begin{aligned} &\begin{aligned}[b] {\varPhi _{T}^{*}[\phi _{2},F_{u}]} :={}&\phi _{2}(u)- \frac{\mu _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{\alpha -1}} \\ &{}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \\ &{}- \frac{\mu _{2}}{p^{ {\alpha \choose 2} +{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta ) } \\ &{}\times \int _{0}^{\frac{T}{p}} \int _{0}^{\frac{x}{p^{-\beta -1}}} \biggl( \frac{T}{p}-qx \biggr) _{p,q}^{\underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}} -qs \biggr)_{p,q}^{\underline{\alpha -1}} \\ &{}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \,d_{p,q}x, \end{aligned} \end{aligned}$$
(3.3)

and the constants \(\mathbf{A}_{\eta }\), \(\mathbf{A}_{T}\), \(\mathbf{B}_{\eta }\), \(\mathbf{B}_{T}\), Ω are defined by (2.5)–(2.9), respectively.

Theorem 3.1

Assume that\(F:I_{p,q}^{T}\times {\mathbb{R}}\times {\mathbb{R}}\times { \mathbb{R}} \rightarrow {\mathbb{R}}\)is continuous, \(\varphi : I_{p,q}^{T}\times I_{p,q}^{T}\rightarrow [0,\infty )\)is continuous with\(\varphi _{0}=\max \{\varphi (t,s):(t,s)\in I_{p,q}^{T}\times I_{p,q}^{T} \}\). Suppose that the following conditions hold:

\((H_{1})\):

There exist constants\(\ell _{1},\ell _{2},\ell _{3}>0\)such that, for each\(t\in I^{T}_{p,q}\)and\(u_{i},v_{i}\in {\mathbb{R}}\), \(i=1,2,3\),

$$\begin{aligned} \bigl\vert F [t,u_{1},u_{2},u_{3} ]-F [ t,v_{1},v_{2},v_{3} ] \bigr\vert \leq &\ell _{1} \vert u_{1}-v_{1} \vert +\ell _{2} \vert u_{2}-v_{2} \vert +\ell _{3} \vert u_{3}-v_{3} \vert . \end{aligned}$$
\((H_{2})\):

There exist constants\(\omega _{1},\omega _{2}>0\)such that, for each\(u, v\in {\mathcal{C}}\),

$$ \bigl\vert \phi _{1}(u)-\phi _{1}(v) \bigr\vert \leq \omega _{1} \Vert u-v \Vert _{\mathcal{C}} \quad \textit{and}\quad \bigl\vert \phi _{2}(u)-\phi _{2}(v) \bigr\vert \leq \omega _{2} \Vert u-v \Vert _{ \mathcal{C}}. $$
\((H_{3})\):

\(\mathcal{X} := ( {\mathcal{L}}+ \ell _{3} ) \varTheta + \omega _{1}\varUpsilon _{T}+\omega _{2}\varUpsilon _{\eta } < 1\),

where

$$\begin{aligned}& {\mathcal{L}} :=\ell _{1}+\ell _{2}\varphi _{0} \frac{ ( \frac{T}{p} )^{\gamma }}{\varGamma _{p,q}(\gamma +1)}, \end{aligned}$$
(3.4)
$$\begin{aligned}& \varTheta := \frac{ ( \frac{T}{p} )^{\alpha}}{\varGamma _{p,q}(\alpha +1)}+{ \mathcal{O}}_{1}\varUpsilon _{T} + {\mathcal{O}}_{2}\varUpsilon _{\eta }. \end{aligned}$$
(3.5)
$$\begin{aligned}& \varUpsilon _{T}:= \frac{1}{ \vert \varOmega \vert } \biggl[ \vert \mathbf{B}_{T} \vert \biggl( \frac{T}{p} \biggr)^{\alpha -1} + \vert \mathbf{A}_{T} \vert \biggl( \frac{T}{p} \biggr)^{\alpha -2} \biggr] , \end{aligned}$$
(3.6)
$$\begin{aligned}& \varUpsilon _{\eta }:=\frac{1}{ \vert \varOmega \vert } \biggl[ \vert \mathbf{B}_{\eta } \vert \biggl( \frac{T}{p} \biggr)^{\alpha -1} + \vert \mathbf{A}_{ \eta } \vert \biggl( \frac{T}{p} \biggr)^{\alpha -2} \biggr], \end{aligned}$$
(3.7)
$$\begin{aligned}& {\mathcal{O}}_{1}:=\frac{\lambda _{1}\eta ^{\alpha}}{\varGamma _{p,q}(\alpha +1)} + \frac{\lambda _{2}\eta ^{\alpha -\beta }}{\varGamma _{p,q}(\alpha -\beta +1)}, \end{aligned}$$
(3.8)
$$\begin{aligned}& {\mathcal{O}}_{2}:=\frac{\lambda _{1} ( \frac{T}{p} )^{\alpha}}{\varGamma _{p,q}(\alpha +1)} + \frac{\lambda _{2} ( \frac{T}{p} )^{\alpha -\beta }}{\varGamma _{p,q}(\alpha -\beta +1)}. \end{aligned}$$
(3.9)

Then problem (1.1) has a unique solution in\(I^{T}_{p,q}\).

Proof

For each \(t\in I_{p,q}^{T}\) and \(u,v\in {\mathcal{C}}\), we have

$$\begin{aligned} \bigl\vert \varPsi ^{\gamma }_{p,q}u-\varPsi ^{\gamma }_{p,q}v \bigr\vert &\leq \frac{\varphi _{0}}{p^{{\gamma \choose 2}}\varGamma _{p,q}(\gamma )} \int _{0}^{t} ( t-qs ) ^{ \underline{\gamma -1}}_{p,q} \biggl\vert u \biggl( \frac{s}{p^{\gamma -1}} \biggr) - v \biggl( \frac{s}{p^{\gamma -1}} \biggr) \biggr\vert \,d_{p,q}s. \\ &\leq \frac{\varphi _{0}}{p^{{\gamma \choose 2}}\varGamma _{p,q}(\gamma )} \vert u-v \vert \int _{0}^{\frac{T}{p}} \biggl( \frac{T}{p}-qs \biggr) ^{\underline{\gamma -1}}_{p,q} \,d_{p,q}s. \\ &=\frac{ \varphi _{0} ( \frac{T}{p} )^{\gamma }}{\varGamma _{p,q}(\gamma +1)} \vert u-v \vert . \end{aligned}$$

Denote that

$$ {\mathcal{H}} \vert u-v \vert (t):= \bigl\vert F \bigl[ t,u(t),\varPsi ^{\gamma }_{p,q}u(t),D^{\nu }_{p,q}u(t) \bigr]-F \bigl[ t,v(t),\varPsi ^{\gamma }_{p,q}v(t),D^{\nu }_{p,q}v(t) \bigr] \bigr\vert . $$

Then we obtain

$$\begin{aligned} &\begin{aligned} & \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} -{\varPhi _{\eta }^{*}[ \phi _{1},F_{v}]} \bigr\vert \\ &\quad \leq \bigl\vert \phi _{1}(u)-\phi _{1}(v) \bigr\vert + \frac{\lambda _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \eta } (\eta -qs )_{p,q}^{\underline{\alpha -1}} { \mathcal{H}} \vert u-v \vert \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &\qquad {}+\frac{\lambda _{2}}{p^{ {\alpha \choose 2}+ {-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta )} \int _{0}^{\eta } \int _{0}^{\frac{x}{p^{-\beta -1}}} (\eta -qx )_{p,q}^{\underline{-\beta -1}} \biggl( \frac{x}{p^{-\beta -1}}-qs \biggr)_{p,q}^{\underline{\alpha -1}}, \end{aligned} \\ &{\mathcal{H}} \vert u-v \vert \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \,d_{p,q}x \\ &\quad \leq \omega _{1} \Vert u-v \Vert _{\mathcal{C}} + \bigl( \ell _{1} \vert u-v \vert +\ell _{2} \bigl\vert \varPsi ^{\gamma }_{p,q}u-\varPsi ^{\gamma }_{p,q}v \bigr\vert + \ell _{3} \bigl\vert D^{\nu }_{p,q}u-D^{\nu }_{p,q}v \bigr\vert \bigr) \\ &\qquad {}\times \biggl\vert \frac{\lambda _{1}\eta ^{\alpha}}{\varGamma _{p,q}(\alpha +1)} + \frac{\lambda _{2}\eta ^{\alpha -\beta }}{\varGamma _{p,q}(\alpha -\beta +1)} \biggr\vert \\ &\quad \leq \omega _{1} \Vert u-v \Vert _{\mathcal{C}} + \biggl( \biggl[ \ell _{1}+ \ell _{2}\varphi _{0} \frac{ ( \frac{T}{p} )^{\gamma }}{\varGamma _{p,q}(\gamma +1)} \biggr] \vert u-v \vert +\ell _{3} \bigl\vert D^{\nu }_{p,q}u-D^{\nu }_{p,q}v \bigr\vert \biggr) {\mathcal{O}}_{1} \\ &\quad \leq \bigl[ \omega _{1}+ ( {\mathcal{L}}+\ell _{3} ) { \mathcal{O}}_{1} \bigr] \Vert u-v \Vert _{\mathcal{C}}. \end{aligned}$$

Similarly,

$$\begin{aligned} \bigl\vert {\varPhi _{T}^{*}[\phi _{2},F_{u}]} -{\varPhi _{T}^{*}[ \phi _{2},F_{v}]} \bigr\vert \leq \bigl[ \omega _{2}+ ( {\mathcal{L}}+\ell _{3} ) { \mathcal{O}}_{2} \bigr] \Vert u-v \Vert _{\mathcal{C}}. \end{aligned}$$

Next, we have

$$\begin{aligned} &\bigl| ({\mathcal{F}}u) (t)-({\mathcal{F}}v) (t) \bigr| \\ &\quad \leq \frac{1}{ p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha ) } \int _{0}^{ \frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{\alpha -1}} {\mathcal{H}} \vert u-v \vert \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &\qquad {}+\frac{ ( \frac{T}{p} )^{\alpha -1}}{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{B}_{T} \vert \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} -{\varPhi _{\eta }^{*}[ \phi _{1},F_{v}]} \bigr\vert + \vert { \mathbf{B}}_{\eta } \vert \bigl\vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} -{\varPhi _{T}^{*}[ \phi _{2},F_{v}]} \bigr\vert \bigr\} \\ &\qquad {}+\frac{ ( \frac{T}{p} )^{\alpha -2}}{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{A}_{T} \vert \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} -{\varPhi _{\eta }^{*}[ \phi _{1},F_{v}]} \bigr\vert + \vert { \mathbf{A}}_{\eta } \vert \bigl\vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} -{\varPhi _{T}^{*}[ \phi _{2},F_{v}]} \bigr\vert \bigr\} \\ &\quad \leq \biggl[ \frac{({\mathcal{L}}+\ell _{3}) ( \frac{T}{p} )^{\alpha}}{\varGamma _{p,q}(\alpha +1)}+ \frac{ [ \omega _{1}+ ( {\mathcal{L}}+\ell _{3} ) {\mathcal{O}}_{1} ] }{\varOmega } \biggl\{ \vert \mathbf{B}_{T} \vert \biggl( \frac{T}{p} \biggr)^{\alpha -1} + \vert { \mathbf{A}}_{T} \vert \biggl( \frac{T}{p} \biggr)^{\alpha -2} \biggr\} \\ &\qquad {}+ \frac{ [ \omega _{2}+ ( {\mathcal{L}}+\ell _{3} ) {\mathcal{O}}_{2} ] }{ \vert \varOmega \vert } \biggl\{ \vert \mathbf{B}_{\eta } \vert \biggl( \frac{T}{p} \biggr)^{\alpha -1} + \vert { \mathbf{A}}_{\eta } \vert \biggl( \frac{T}{p} \biggr)^{\alpha -2} \biggr\} \biggr] \Vert u-v \Vert _{\mathcal{C}} \\ &\quad =\mathcal{X} \Vert u-v \Vert _{\mathcal{C}}. \end{aligned}$$
(3.10)

Taking fractional \((p,q)\)-difference of order ν for (3.1), we get

$$\begin{aligned} & \bigl(D_{p,q}^{\nu }{\mathcal{F}}u \bigr) (t) \\ &\quad =\frac{1}{p^{ {\alpha \choose 2}+ {-\nu \choose 2}}\varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\nu ) } \int _{0}^{t} \int _{0}^{ \frac{x}{p^{-\nu -1}} } (t-qx )_{p,q}^{ \underline{-\nu -1}} \biggl( \frac{x}{p^{-\nu -1}} -qs \biggr)_{p,q}^{ \underline{\alpha -1}} \\ &\qquad {}\times F \biggl[ \frac{s}{p^{\alpha -1}},u \biggl( \frac{s}{p^{\alpha -1}} \biggr) ,\varPsi ^{\gamma }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr),D^{\nu }_{p,q}u \biggl( \frac{s}{p^{\alpha -1}} \biggr) \biggr] \,d_{p,q}s \,d_{p,q}x \\ &\qquad {}-\frac{1}{\varOmega p^{{-\nu \choose 2}} \varGamma _{p,q}(-\nu )} \bigl\{ \mathbf{B}_{T}{\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} -{\mathbf{B}}_{\eta }{ \varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\} \\ &\qquad {}\times \int _{0}^{t} (t-qs )_{p,q}^{\underline{-\nu -1}} \biggl( \frac{s}{p^{-\nu -1}} \biggr)^{\alpha -1} \,d_{p,q}s \\ &\qquad {}+\frac{1}{\varOmega p^{{-\nu \choose 2}}\varGamma _{p,q}(-\nu )} \bigl\{ \mathbf{A}_{T}{\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} -{\mathbf{A}}_{\eta }{ \varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\} \\ &\qquad {}\times \int _{0}^{t} (t-qs )_{p,q}^{\underline{-\nu -1}} \biggl( \frac{s}{p^{-\nu -1}} \biggr)^{\alpha -2} \,d_{p,q}s. \end{aligned}$$
(3.11)

Similarly, we have

$$\begin{aligned} \bigl\vert \bigl(D_{p,q}^{\nu }{ \mathcal{F}}u \bigr) (t) - \bigl(D_{p,q}^{\nu }{\mathcal{F}}v \bigr) (t) \bigr\vert < \mathcal{X} \Vert u-v \Vert _{\mathcal{C}}. \end{aligned}$$
(3.12)

From (3.10) and (3.12), we obtain

$$ \Vert {\mathcal{F}}u-{\mathcal{F}}v \Vert _{\mathcal{C}}\leq \mathcal{X} \Vert u-v \Vert _{\mathcal{C}}. $$

By \((H_{3})\) we can conclude that \({\mathcal{F}}\) is a contraction. Therefore, by using Banach’s fixed point theorem, \({\mathcal{F}}\) has a fixed point which is a unique solution of problem (1.1) on \(I^{T}_{p,q}\). □

Existence of at least one solution

In this section, we present the existence of a solution to (1.1) by using Schauder’s fixed point theorem.

Lemma 4.1

([57])

(Arzelá–Ascoli theorem) A collection of functions in\(C[a,b]\)with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on\([a,b]\).

Lemma 4.2

([57])

If a set is closed and relatively compact, then it is compact.

Lemma 4.3

([58] (Schauder’s fixed point theorem))

Let\((D,d)\)be a complete metric space, Ube a closed convex subset ofD, and\(T: D\rightarrow D\)be the map such that the set\(Tu:u\in U\)is relatively compact inD. Then the operatorThas at least one fixed point\(u^{*}\in U\): \(Tu^{*}=u^{*}\).

Theorem 4.1

Suppose that\((H_{1})\)and\((H_{3})\)hold. Then problem (1.1) has at least one solution on\(I^{T}_{p,q}\).

Proof

We organize the proof into three steps as follows.

Step I. Verify that \({\mathcal{F}}\) maps bounded sets into bounded sets in \(B_{L} = \{u \in \mathcal{C}: \|u\|_{\mathcal{C}} \leq L\}\). Set \(\max_{t\in I^{T}_{p,q}}|F(t,0,0,0)|=M\), \(\sup_{u\in {\mathcal{C}}} |\phi _{1}(u)|= N_{1}\), \(\sup_{u\in {\mathcal{C}}} |\phi _{2}(u)|= N_{2}\) and choose a constant

$$ L\geq \frac{ M\varTheta +N_{1}\varUpsilon _{T}+N_{2}\varUpsilon _{\eta }}{ 1-(\mathcal{L}+\ell _{3})\varTheta }. $$
(4.1)

Denote that \(|\mathcal{S}(t,u,0) |= |F [t,u(t),\varPsi _{p,q}^{\gamma }u(t),D_{p,q}^{\nu }u(t) ]-F [t,0,0,0 ] |+ |F[t,0,0,0] | \). For each \(t\in I_{p,q}^{T}\) and \(u\in B_{L}\), we obtain

$$\begin{aligned} & \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} \bigr\vert \\ &\quad \leq N_{1}+ \frac{\lambda _{1}}{p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha )} \int _{0}^{ \eta } (\eta -qs )_{p,q}^{\underline{\alpha -1}} \bigl\vert \mathcal{S}(s,u,0) \bigr\vert \,d_{p,q}s + \frac{\lambda _{2}}{p^{{\alpha \choose 2}+{-\beta \choose 2}} \varGamma _{p,q}(\alpha )\varGamma _{p,q}(-\beta )} \\ &\qquad {}\times \int _{0}^{\eta } \int _{0}^{\frac{x}{p^{-\beta -1}}} (\eta -qx )_{p,q}^{\underline{-\beta -1}} \biggl(\frac{x}{p^{-\beta -1}}-qs \biggr)_{p,q}^{\underline{\alpha -1}} \bigl\vert \mathcal{S}(s,u,0) \bigr\vert \,d_{p,q}s \,d_{p,q}x \\ &\quad \leq N_{1}+ \biggl( \biggl[ \ell _{1}+\ell _{2}\varphi _{0} \frac{ ( \frac{T}{p} )^{\gamma }}{\varGamma _{p,q}(\gamma +1)} \biggr] \vert u \vert +\ell _{3} \bigl\vert D^{\nu }_{p,q}u \bigr\vert +M \biggr) \mathcal{O}_{1} \\ &\quad \leq N_{1}+M\mathcal{O}_{1}+ ( {\mathcal{L}}+ \ell _{3} ) \mathcal{O}_{1} \Vert u \Vert _{\mathcal{C}} \\ &\quad \leq N_{1}+ \bigl[ M+ ( {\mathcal{L}}+\ell _{3} )L \bigr] \mathcal{O}_{1}. \end{aligned}$$
(4.2)

Similarly,

$$\begin{aligned} \bigl\vert {\varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\vert \leq N_{2}+ \bigl[ M+ ( {\mathcal{L}}+\ell _{3} )L \bigr] \mathcal{O}_{2}. \end{aligned}$$
(4.3)

From (4.2)–(4.3), we find that

$$\begin{aligned} | ({\mathcal{F}}u) (t)| \leq {}&\frac{1}{ p^{{\alpha \choose 2}}\varGamma _{p,q}(\alpha ) } \int _{0}^{\frac{T}{p}} \biggl(\frac{T}{p}-qs \biggr)_{p,q}^{ \underline{\alpha -1}} \bigl\vert \mathcal{S}(t,u,0) \bigr\vert \biggl( \frac{s}{p^{\alpha -1}} \biggr) \,d_{p,q}s \\ &{}+\frac{ ( \frac{T}{p} )^{\alpha -1}}{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{B}_{T} \vert \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} \bigr\vert + \vert { \mathbf{B}}_{\eta } \vert \bigl\vert { \varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\vert \bigr\} \\ &{}+\frac{ ( \frac{T}{p} )^{\alpha -2}}{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{A}_{T} \vert \bigl\vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} \bigr\vert + \vert { \mathbf{A}}_{\eta } \vert \bigl\vert { \varPhi _{T}^{*}[\phi _{2},F_{u}]} \bigr\vert \bigr\} \\ \leq {} & \varTheta \bigl[ L ( {\mathcal{L}}+\ell _{3} )+M \bigr] +N_{1} \varUpsilon _{T}+N_{2}\varUpsilon _{\eta } \\ \leq {}&L. \end{aligned}$$
(4.4)

In addition, we obtain

$$\begin{aligned} \bigl\vert \bigl( D^{\nu }_{p,q}{ \mathcal{F}}u \bigr) (t) \bigr\vert < L. \end{aligned}$$
(4.5)

Therefore, \(\|{\mathcal{F}}u\|_{\mathcal{C}}\leq L\), which implies that \({\mathcal{F}}\) is uniformly bounded.

Step II. The operator \({\mathcal{F}}\) is continuous on \(B_{L}\) because of the continuity of F.

Step III. We examine that \({\mathcal{F}}\) is equicontinuous on \(B_{L}\). For any \(t_{1},t_{2}\in I^{T}_{p,q}\) with \(t_{1}< t_{2}\), we have

$$\begin{aligned} | ({\mathcal{F}}u) (t_{2})- ({\mathcal{F}}u) (t_{1}) | \leq {}&\frac{ \Vert F \Vert }{\varGamma _{p,q}(\alpha +1)} \bigl\vert t_{2}^{\alpha}-t_{1}^{\alpha} \bigr\vert \\ &{}+\frac{ \vert t_{2}^{\alpha -1} -t_{1}^{\alpha -1} \vert }{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{B}_{T} \vert { \varPhi _{\eta }^{*}[\phi _{1},F_{u}]} + \vert { \mathbf{B}}_{\eta } \vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} \bigr\} \\ &{}+\frac{ \vert t_{2}^{\alpha -2} -t_{1}^{\alpha -2} \vert }{ \vert \varOmega \vert } \bigl\{ \vert \mathbf{A}_{T} \vert { \varPhi _{\eta }^{*}[\phi _{1},F_{u}]} + \vert { \mathbf{A}}_{\eta } \vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} \bigr\} \end{aligned}$$
(4.6)

and

$$\begin{aligned} & \bigl\vert \bigl(D_{p,q}^{\nu }{\mathcal{F}}u \bigr) (t_{2}) - \bigl(D_{p,q}^{\nu }{\mathcal{F}}u \bigr) (t_{1}) \bigr\vert \\ &\quad \leq\frac{ \Vert F \Vert }{\varGamma _{p,q}(\alpha -\nu +1)} \bigl\vert t_{2}^{ \alpha -\nu } -t_{1}^{\alpha -\nu } \bigr\vert \\ &\qquad {}+\frac{\varGamma _{p,q}(\alpha )}{ \vert \varOmega \vert \varGamma _{p,q}(\alpha -\nu )} \bigl\{ \vert \mathbf{B}_{T} \vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} + \vert { \mathbf{B}}_{\eta } \vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} \bigr\} \bigl\vert t_{2}^{ \alpha -\nu -1}-t_{1}^{\alpha -\nu -1} \bigr\vert \\ &\qquad {}+\frac{\varGamma _{p,q}(\alpha -1)}{ \vert \varOmega \vert \varGamma _{p,q}(\alpha -\nu -1)} \bigl\{ \vert \mathbf{A}_{T} \vert {\varPhi _{\eta }^{*}[\phi _{1},F_{u}]} + \vert { \mathbf{A}}_{\eta } \vert {\varPhi _{T}^{*}[ \phi _{2},F_{u}]} \bigr\} \bigl\vert t_{2}^{ \alpha -\nu -2}-t_{1}^{\alpha -\nu -2} \bigr\vert . \end{aligned}$$
(4.7)

Since the right-hand side of (4.6) and (4.7) tends to be zero when \(|t_{2}-t_{1}|\rightarrow 0\), \({\mathcal{F}}\) is relatively compact on \(B_{L}\).

This implies that \({\mathcal{F}}(B_{L})\) is an equicontinuous set. From Steps I to III together with the Arzelá–Ascoli theorem, we see that \(\mathcal{F}:{\mathcal{C}}\rightarrow {\mathcal{C}}\) is completely continuous. By Schauder’s fixed point theorem, we can conclude that problem (1.1) has at least one solution. □

An example

Consider the following fractional \((p,q)\)-integrodifference equation:

$$\begin{aligned} D^{\frac{4}{3}}_{\frac{2}{3},\frac{1}{2}}u(t) =&\frac{1}{ ( 2000e^{3}+t^{3} )(1+ \vert u(t) \vert )} \bigl[ e^{-2t} \bigl( u^{2}+2 \vert u \vert \bigr) + e^{-(2\pi +\cos ^{2}\pi t)} \bigl\vert \varPsi _{\frac{2}{3},\frac{1}{2}}^{\frac{1}{2}}u(t) \bigr\vert \\ &{}+ e^{-(2+\sin ^{2}\pi t)} \bigl\vert D^{\frac{2}{5}}_{\frac{2}{3}, \frac{1}{2}}u(t) \bigr\vert \bigr],\quad t\in I^{10}_{\frac{2}{3}, \frac{1}{2}}= \biggl\lbrace \frac{10 ( \frac{1}{2} )^{k}}{ ( \frac{2}{3} )^{k+1}}:k \in {\mathbb{N}}_{0} \biggr\rbrace \cup \{0\} \end{aligned}$$

with the nonlocal Robin boundary condition

$$\begin{aligned}& \frac{1}{20e} u \biggl(\frac{1215}{256} \biggr)+200e D_{\frac{2}{3}, \frac{1}{2}}^{\frac{3}{4}}u \biggl(\frac{1215}{256} \biggr)= \sum _{i=0}^{\infty } \frac{C_{i} \vert u(t_{i}) \vert }{1+ \vert u(t_{i}) \vert },\quad t_{i} = 10 \frac{ ( \frac{1}{2} )^{i} }{ ( \frac{2}{3} )^{i+1} }, \\& 100\pi u (15 )+\frac{1}{10 \pi } D_{\frac{2}{3},\frac{1}{2}}^{ \frac{3}{4}}u (15 )= \sum_{i=0}^{\infty } \frac{D_{i} \vert u(t_{i}) \vert }{1+ \vert u(t_{i}) \vert },\quad t_{i} = 10 \frac{ ( \frac{1}{2} )^{i} }{ ( \frac{2}{3} )^{i+1} }, \end{aligned}$$

where \(\varphi (t,s)=\frac{e^{-|s-t|}}{(t+20)^{3}}\) and \(C_{i}\), \(D_{i}\) are given constants with \(\frac{1}{500}\leq \sum_{i=0}^{\infty }C_{i}\leq \frac{\pi }{1000}\) and \(\frac{1}{1000}\leq \sum_{i=0}^{\infty }D_{i}\leq \frac{\pi }{2000}\).

Letting \(\alpha =\frac{4}{3}\), \(\beta =\frac{3}{4}\), \(\gamma =\frac{1}{2}\), \(\nu = \frac{2}{5}\), \(p=\frac{2}{3}\), \(q=\frac{1}{2}\), \(T=10\), \(\eta =10 \frac{ ( \frac{1}{2} )^{4} }{ ( \frac{2}{3} )^{5} }= \frac{1215}{256}\), \(\lambda _{1}=\frac{1}{20e} \), \(\lambda _{2}=200e\), \(\mu _{1}=100\pi \), \(\mu _{2}=\frac{1}{10 \pi }\), \(\phi _{1}(u)=\sum_{i=0}^{\infty } \frac{C_{i}|u(t_{i})|}{1+|u(t_{i})|}\), \(\phi _{2}=\sum_{i=0}^{\infty } \frac{D_{i}|u(t_{i})|}{1+|u(t_{i})|}\) and \(F [ t,u(t),\varPsi ^{\gamma }_{p,q}u(t), D^{\nu }_{p,q}u(t) ] = \frac{1}{ ( 2000e^{3}+t^{3} )(1+|u(t)|)} [ e^{-2t} ( u^{2}+2|u| ) + e^{-(2\pi +\cos ^{2}\pi t)} \vert \varPsi _{ \frac{2}{3},\frac{1}{2}}^{\frac{1}{2}}u(t) \vert + e^{-(2+\sin ^{2} \pi t)} \vert D^{\frac{2}{5}}_{\frac{2}{3},\frac{1}{2}}u(t) \vert ]\), we find that

$$\begin{aligned}& \vert {\mathbf{A}}_{\eta } \vert =574.6570,\qquad \vert { \mathbf{A}}_{T} \vert =-23.8344,\qquad \vert { \mathbf{B}}_{\eta } \vert =774.8145,\qquad \vert { \mathbf{B}}_{T} \vert =51.6518,\\& \varphi _{0}=0.000125, \quad \mbox{and} \quad \vert \varOmega \vert =-48\mbox{,}149.3072. \end{aligned}$$

For all \(t\in I^{10}_{\frac{2}{3},\frac{1}{2}}\) and \(u, v\in {\mathbb{R}}\), we have

$$\begin{aligned} \begin{aligned} & \bigl\vert F \bigl[ t,u,\varPsi ^{\gamma }_{p,q}u,D^{\nu }_{p,q}u \bigr]- F \bigl[ t,v,\varPsi ^{\gamma }_{p,q}v,D^{\nu }_{p,q}v \bigr] \bigr\vert \\ &\quad \leq \frac{1}{2000e^{2} } \vert u-v \vert +\frac{1}{2000e^{2\pi +2} } \bigl\vert \varPsi ^{\gamma }_{p,q}u-\varPsi ^{\gamma }_{p,q}v \bigr\vert + \frac{1}{2000e^{4} } \bigl\vert D^{\nu }_{p,q}u-D^{\nu }_{p,q}v \bigr\vert . \end{aligned} \end{aligned}$$

Thus, \((H_{1})\) holds with \(\ell _{1}=6.767\times 10^{-5}\), \(\ell _{2}=1.264\times 10^{-7}\), and \(\ell _{3}=9.158\times 10^{-6}\). For all \(u, v\in {\mathcal{C}}\),

$$\begin{aligned}& \bigl\vert \phi _{1}(u)-\phi _{1}(v) \bigr\vert \leq \frac{\pi }{1000} \Vert u-v \Vert _{\mathcal{C}}, \\& \bigl\vert \phi _{2}(u)-\phi _{2}(v) \bigr\vert \leq \frac{e}{2000} \Vert u-v \Vert _{\mathcal{C}}. \end{aligned}$$

So, \((H_{2})\) holds with \(\omega _{1}= 0.003142\) and \(\omega _{2}=0.001571\). Since

$$\begin{aligned}& {\mathcal{L}} = 0.0000677,\qquad \mathcal{O}_{1}=1415.89969,\qquad \mathcal{O}_{2}=2770.8547,\\& \varUpsilon _{T}=0.005291, \qquad \varUpsilon _{\eta }=0.003183,\quad \mbox{and}\quad \varTheta =51 \mbox{,}3459, \end{aligned}$$

\((H_{3})\) holds with

$$ \mathcal{X} \approx 0.00397< 1. $$

Hence, by Theorem 3.1 this problem has a unique solution. Moreover, by Theorem 4.1 this problem has at least one solution.

References

  1. 1.

    Jackson, F.H.: On q-difference equations. Am. J. Math. 32, 305–314 (1910)

    MATH  Google Scholar 

  2. 2.

    Jackson, F.H.: On q-difference integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  3. 3.

    Carmichael, R.D.: The general theory of linear q-difference equations. Am. J. Math. 34, 147–168 (1912)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Mason, T.E.: On properties of the solutions of linear q-difference equations with entire function coefficients. Am. J. Math. 37, 439–444 (1915)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Adams, C.R.: On the linear ordinary q-difference equation. Am. Math. Ser. II 30, 195–205 (1929)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Trjitzinsky, W.J.: Analytic theory of linear q-difference equations. Acta Math. 62(1), 167–226 (1933)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    Google Scholar 

  8. 8.

    Ernst, T.: A new notation for q-calculus and a new q-Taylor formula. U.U.D.M. Report 1999:25, ISSN 1101-3591, Department of Mathematics, Uppsala University, (1999)

  9. 9.

    Floreanini, R., Vinet, L.: q-gamma and q-beta functions in quantum algebra representation theory. J. Comput. Appl. Math. 68, 57–68 (1996)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Finkelstein, R.J.: Symmetry group of the hydrogen atom. J. Math. Phys. 8(3), 443–449 (1967)

    MathSciNet  Google Scholar 

  11. 11.

    Gavrilik, A.M.: q-Serre relations in \(U_{q}(u_{n})\) and q-deformed meson mass sum rules. J. Phys. A, Math. Gen. 27(3), 91–94 (1994)

    MATH  Google Scholar 

  12. 12.

    Finkelstein, R.J.: q-Gauge theory. Int. J. Mod. Phys. A 11(4), 733–746 (1996)

    MATH  Google Scholar 

  13. 13.

    Kaniadakis, G., Lavagno, A., Quarati, P.: Kinetic model for q-deformed bosons and fermions. Phys. Lett. A 227(3–4), 227–231 (1997)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ilinski, K.N., Kalinin, G.V., Stepanenko, A.S.: q-functional field theory for particles with exotic statistics. Phys. Lett. A 232(6), 399–408 (1997)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Finkelstein, R.J.: q-field theory. Lett. Math. Phys. 34(2), 169–176 (1995)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Chan, F.L., Finkelstein, R.J.: q-deformation of the Coulomb problem. J. Math. Phys. 35(7), 3273–3284 (1994)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Cadavid, A.C., Finkelstein, R.J.: The q-Coulomb problem in configuration space. J. Math. Phys. 37(8), 3675–3683 (1996)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Finkelstein, R.J.: The q-Coulomb problem. J. Math. Phys. 37(6), 2628–2636 (1996)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Feigenbaum, J., Freund, P.G.: A q-deformation of the Coulomb problem. J. Math. Phys. 37(4), 1602–1616 (1996)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Cheng, H.: Canonical Quantization of Yang–Mills Theories, Perspectives in Mathematical Physics. International Press, Somerville (1996)

    Google Scholar 

  21. 21.

    Finkelstein, R.J.: q-gravity. Lett. Math. Phys. 38(1), 53–62 (1996)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kamata, M., Nakamula, A.: One-parameter family of selfdual solutions in classical Yang–Mills theory. Phys. Lett. B 463(2–4), 257–262 (1999)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Negadi, T., Kibler, M.: A q-deformed Aufbau Prinzip. J. Phys. A, Math. Gen. 25(4), 157–160 (1992)

    MathSciNet  Google Scholar 

  24. 24.

    Marinova, L.P., Raychev, P.P., Maruani, J.: Molecular backbending in AgH and its description in terms of q-algebras. Mol. Phys. 82(6), 1115–1129 (1994)

    MathSciNet  Google Scholar 

  25. 25.

    Monteiro, M.R., Rodrigues, L.M.C.S., Wulck, S.: Quantum algebraic nature of the photon spectrum in 4He. Phys. Rev. Lett. 76(7), 1098–1100 (1996)

    Google Scholar 

  26. 26.

    Siegel, W.: Introduction to String Field Theory. Advanced Series in Mathematical Physics, vol. 8. World Scientific Publishing, Teaneck (1988)

    Google Scholar 

  27. 27.

    Chakrabarti, R., Jagannathan, R.: A \((p,q)\)-oscillator realization of two-parameter quantum algebras. J. Phys. A, Math. Gen. 24(24), 5683–5701 (1991)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Jagannathan, R., Rao, K.S.: Two-parameter quantum algebras, twin-basic number, and associated generalized hypergeometric series. Differ. Equ. Appl. 2006, 27 (2006)

    Google Scholar 

  29. 29.

    Sahai, V., Yadav, S.: Representations of two parameter quantum algebras and \((p,q)\)-special functions. J. Math. Anal. Appl. 335(1), 268–279 (2007)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Burban, I.: Two-parameter deformation of the oscillator algebra and \((p,q)\)-analog of two-dimensional conformal field theory. J. Nonlinear Math. Phys. 2(3–4), 384–391 (1995)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Sadjang, P.N.: On the fundamental theorem of \((p,q)\)-calculus and some \((p,q)\)-Taylor formulas. Results Math. 73, 39 (2018)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Hounkonnou, M.N., Kyemba, J.D.: \(R(p,q)\)-calculus: differentiation and integration. SUT J. Math. 49(2), 145–167 (2013)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Mursaleen, M., Ansari, K.J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Araci, S., Duran, U.G., Acikgoz, M., Srivastava, H.M.: A certain \((p,q)\)-derivative operator and associated divided differences. J. Inequal. Appl. 2016, 301 (2016)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Duran, U.: Post Quantum Calculus, [Master Thesis]. University of Gaziantep, (2016)

  36. 36.

    Kamsrisuk, N., Promsakon, C., Ntouyas, S.K., Tariboon, J.: Nonlocal boundary value problems for \((p,q)\)-difference equations. Differ. Equ. Appl. 10(2), 183–195 (2018)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Promsakon, C., Kamsrisuk, N., Ntouyas, S.K., Tariboon, J.: On the second-order quantum \((p,q)\)-difference equations with separated boundary conditions. Adv. Math. Phys. 2018, Article ID 9089865 (2018)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Milovanovic, G.V., Gupta, V., Malik, N.: \((p,q)\)-Beta functions and applications in approximation. Bol. Soc. Mat. Mex. 24, 219–237 (2018)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Cheng, W.T., Zhang, W.H., Cai, Q.B.: \((p,q)\)-gamma operators which preserve \(x^{2}\). J. Inequal. Appl. 2019, 108 (2019)

    Google Scholar 

  40. 40.

    Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015) [Corrigendum: Appl. Math. Comput. 269, 744–746 (2015)]

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Mursaleen, M., Nasiruzzaman, M., Khan, A., Ansari, K.J.: Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p,q)\)-integers. Filomat 30(3), 639–648 (2016)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Mursaleen, M., Khan, F., Khan, A.: Approximation by \((p,q)\)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory 10(8), 1725–1740 (2016)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Rahman, S., Mursaleen, M., Alkhaldi, A.H.: Convergence of iterates of q-Bernstein and \((p,q)\)-Bernstein operators and the Kelisky–Rivlin type theorem. Filomat 32(12), 4351–4364 (2018)

    MathSciNet  Google Scholar 

  44. 44.

    Jebreen, H.B., Mursaleen, M., Ahasan, M.: On the convergence of Lupaş \((p,q)\))-Bernstein operators via contraction principle. J. Inequal. Appl. 2019, 34 (2019)

    MathSciNet  Google Scholar 

  45. 45.

    Nasiruzzaman, M., Mukheimer, A., Mursaleen, M.: Some Opial-type integral inequalities via \((p,q)\)-calculus. J. Inequal. Appl. 2019, 5 (2019)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Khan, A., Sharma, V.: Statistical approximation by \((p,q)\)-analogue of Bernstein–Stancu operators. Azerb. J. Math. 8(2), 100–121 (2018)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Khan, K., Lobiyal, D.K.: Beézier curves based on Lupaş \((p,q)\)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 317, 458–477 (2017)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Agarwal, R.P.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philol. Soc. 66, 365–370 (1969)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Al-Salam, W.A.: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1966)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Díaz, J.B., Osler, T.J.: Differences of fractional order. Math. Compet. 28, 185–202 (1974)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Brikshavana, T., Sitthiwirattham, T.: On fractional Hahn calculus. Adv. Differ. Equ. 2017, 354 (2017)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Patanarapeelert, N., Sitthiwirattham, T.: On fractional symmetric Hahn calculus. Mathematics 7, 873 (2019)

    Google Scholar 

  53. 53.

    Soontharanon, J., Sitthiwirattham, T.: On fractional \((p,q)\)-calculus. Adv. Differ. Equ. 2020, 35 (2020)

    Google Scholar 

  54. 54.

    Promsakon, C., Kamsrisuk, N., Ntouyas, S.K., Tariboon, J.: On the second-order \((p,q)\)-difference equations with separated boundary conditions. Adv. Math. Phys. 2018, 9089865 (2018)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Promsakon, C., Kamsrisuk, N., Ntouyas, S.K., Tariboon, J.: Nonlocal boundary value problems for \((p,q)\)-difference equations. Differ. Equ. Appl. 10, 183–195 (2018)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Nuntigrangjana, T., Putjuso, S., Ntouyas, S.K., Tariboon, J.: Impulsive quantum \((p,q)\)-difference equations. Adv. Differ. Equ. 2020, 98 (2020)

    MathSciNet  Google Scholar 

  57. 57.

    Griffel, D.H.: Applied Functional Analysis. Ellis Horwood Publishers, Chichester (1981)

    Google Scholar 

  58. 58.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cone. Academic Press, Orlando (1988)

    Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous referee for very helpful suggestions and comments which led to improvements of our original manuscript.

Availability of data and materials

Not applicable.

Funding

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-026.

Author information

Affiliations

Authors

Contributions

The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript.

Corresponding author

Correspondence to Thanin Sitthiwirattham.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Soontharanon, J., Sitthiwirattham, T. Existence results of nonlocal Robin boundary value problems for fractional \((p,q)\)-integrodifference equations. Adv Differ Equ 2020, 342 (2020). https://doi.org/10.1186/s13662-020-02806-7

Download citation

MSC

  • 39A10
  • 39A13
  • 39A70

Keywords

  • Fractional \((p,q)\)-integral
  • Fractional \((p,q)\)-difference
  • Robin boundary value problems
  • Existence