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.
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1 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 [3–9] and their references. The knowledge of q-calculus was used in physical problems, see [10–27] 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 [28–39]. 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. [40–47].
Then, the study of fractional quantum calculus was initiated [48–50]. 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 [54–56]. 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
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
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.
2 Preliminaries
In this section, we recall some basic definitions, notations, and lemmas. Letting \(0< q< p\leq 1\), we define the notations
The \((p,q)\)-forward jump and the \((p,q)\)-backward jump operators are defined as
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
The \((p,q)\)-analogue of the power function \((a-b)_{p,q}^{\underline{n}}\) with \(n\in \mathbb{N}_{0}\) is given by
For \(\alpha \in \mathbb{R}\), we define a general form:
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
respectively.
Lemma 2.1
([53])
For\(\alpha ,\beta ,\gamma ,\lambda \in {\mathbb{R}}\),
Lemma 2.2
([53])
For\(m,n\in {\mathbb{N}}_{0}\), \(\alpha \in \mathbb{R}\), and\(0< q< p\leq 1\),
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
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:
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
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
where
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
The relations between \((p,q)\)-difference and \((p,q)\)-integral operators are given by
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:
Lemma 2.6
([31], Fundamental theorem of \((p,q)\)-calculus)
Letting\(f:I\rightarrow \mathbb{R}\)be continuous at 0 and
thenFis continuous at 0 and\(D_{p,q}F(x)\)exists for every\(x\in I\)where
Conversely,
Lemma 2.7
([53], Leibniz formula of \((p,q)\)-calculus)
Letting\(f:I_{p,q}^{T}\times I_{p,q}^{T}\rightarrow \mathbb{R}\),
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
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
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\(f:I_{p,q}^{T}\rightarrow {\mathbb{R}}\),
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,
Lemma 2.10
([53])
Let\(\alpha ,\beta >0\), \(0< q< p\leq 1\). Then
Lemma 2.11
Let\(\alpha ,\beta >0\), \(0< q< p\leq 1\), and\(n \in \mathbb{Z}\). Then
Proof
From Lemma 2.10(a) and the definition of \((p,q)\)-beta function, we have
For \(n \in \mathbb{Z}\), we have
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)
has the unique solution
where the functionals\({\varPhi _{\eta }[\phi _{1},h]}\), \({\varPhi _{T}[\phi _{2},h]}\)are defined by
and the constants\(\mathbf{A}_{\eta }\), \(\mathbf{A}_{T}\), \(\mathbf{B}_{\eta }\), \(\mathbf{B}_{T}\), andΩare defined by
Proof
To obtain the solution, we first take a fractional \((p,q)\)-integral of order α for (2.1). Then we have
Next, we take fractional p, q-difference of order β for (2.10) to get
Substituting \(t=\eta \) into (2.10) and (2.11) and employing the first condition of (2.1), we have
Taking \(t=T\) into (2.10) and (2.11) and employing the second condition of (2.1), we have
Solving \(\text{(2.12)}--\text{(2.13)}\), we find that
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. □
3 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
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
where the functionals \({\varPhi _{\eta }^{*}[\phi _{1},F_{u}]}\), \({\varPhi _{T}^{*}[\phi _{2},F_{u}]}\) are defined by
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
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
Denote that
Then we obtain
Similarly,
Next, we have
Taking fractional \((p,q)\)-difference of order ν for (3.1), we get
Similarly, we have
From (3.10) and (3.12), we obtain
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 \(I^{T}_{p,q}\). □
4 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
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
Similarly,
From (4.2)–(4.3), we find that
In addition, we obtain
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
and
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. □
5 An example
Consider the following fractional \((p,q)\)-integrodifference equation:
with the nonlocal Robin boundary condition
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
For all \(t\in I^{10}_{\frac{2}{3},\frac{1}{2}}\) and \(u, v\in {\mathbb{R}}\), we have
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 \(u, v\in {\mathcal{C}}\),
So, \((H_{2})\) holds with \(\omega _{1}= 0.003142\) and \(\omega _{2}=0.001571\). Since
\((H_{3})\) holds with
Hence, by Theorem 3.1 this problem has a unique solution. Moreover, by Theorem 4.1 this problem has at least one solution.
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This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-026.
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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
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DOI: https://doi.org/10.1186/s13662-020-02806-7