Abstract
In this paper, we propose a nonlocal fractional sum-difference boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator. The problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums. The existence and uniqueness result of the problem is studied by using the Banach fixed point theorem.
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1 Introduction
Discrete fractional calculus and fractional difference equations have been widely studied. Goodrich and Peterson gave some useful basic definitions and properties of fractional difference calculus in the book [1]. Discrete fractional calculus can be applied in queuing problems, economics, logistic map, and electrical networks, see [2–4]. The extension of discrete fractional calculus has helped to build up some of the basic theory in this area, see [5–32] and the references cited therein.
The boundary value problem for fractional differential equations and the system of equations with p-Laplacian operator were presented in [33–39] and [40–44], respectively. Particularly, the boundary value problem for fractional difference equations with p-Laplacian operator was presented in [45–47]. In addition, the existence results of systems of fractional boundary value problems were presented in [48–55].
We observe that the boundary value problem of a coupled system of nonlinear fractional difference equations with p-Laplacian operator has not been studied. This result is the motivation for this research. In this paper, we aim to study the coupled system of nonlinear fractional sum-difference equations with p-Laplacian operator
with the nonlocal fractional sum and fractional difference boundary conditions
where \(t\in \mathbb{N}_{0,T}:=\{0,1,\ldots,T\}\), \(\alpha _{i},\beta _{i},\gamma _{i},\omega _{i},\theta _{i}\in (0,1)\), \(\alpha _{i}+\beta _{i}\in (1,2]\), \(\lambda _{i}>0\), \(\eta _{i}\in { \mathbb{N}}_{\alpha _{i}+\beta _{i}-1,T+\alpha _{i}+\beta _{i}-1}\), \(F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\times \mathbb{R}^{3}, \mathbb{R} )\), \(g_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R}^{+} )\), and for \(\varphi _{i} : \mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\rightarrow [0,\infty )\),
for \(i\in \{1,2\}\). For \(p>1\), the p-Laplacian operator is defined as \(\phi _{p}(x)=|x|^{p-2}x\), where \(\phi _{p}\) is invertible and its inverse operator is \(\phi _{q}\), where \(q>1\) is a constant such that \(\frac{1}{p}+\frac{1}{q}=1\).
Our plan is as follows. In Sect. 2, we recall some basic knowledge and convert (1.1)–(1.2) to an equivalent summation equation and find its solution. In Sect. 3, we prove existence and uniqueness of the solution of boundary value problem (1.1)–(1.2) by using the Banach fixed point theorem. Some examples to illustrate our result are presented in the last section.
2 Preliminaries
Notations, definitions, and lemmas which are used in the main results are given as follows.
Definition 2.1
The generalized falling function is defined by \(t^{\underline{\alpha }}:= \frac{\varGamma (t+1)}{\varGamma (t+1-\alpha )}\) for any t and α for which the right-hand side is defined. If \(t+1-\alpha \) is a pole of the gamma function and \(t+1\) is not a pole, then \(t^{\underline{\alpha }}=0\).
Lemma 2.1
([5])
Assume that the following factorial functions are well defined:
-
(i)
\((t-\mu ) t^{\underline{\mu }}=t^{\underline{\mu +1}}\), where\(\mu \in \mathbb{R}\).
-
(ii)
If\(t\leq r\), then\(t^{\underline{\alpha }}\leq r^{\underline{\alpha }}\)for any\(\alpha >0\).
Definition 2.2
Let \(\alpha >0\) and f be defined on \(\mathbb{N}_{a}\), the α-order fractional sum of f is defined by
where \(t\in \mathbb{N}_{a+\alpha }\) and \(\sigma (s)=s+1\).
Definition 2.3
For \(\alpha >0\) and f defined on \(\mathbb{N}_{a}\), the α-order Riemann–Liouville fractional difference of f is defined by
The α-order Caputo fractional difference of f is defined by
where \(t\in \mathbb{N}_{a+N-\alpha }\) and \(N \in \mathbb{N}\) is chosen so that \(0\leq N-1<\alpha < N\). If \(\alpha =N\), then \(\Delta ^{\alpha }f(t)=\Delta ^{\alpha }_{C} f(t)=\Delta ^{N} f(t)\).
Lemma 2.2
([7])
Let\(0\leq N-1<\alpha \leq N\). Then
for some\(C_{i}\in \mathbb{R}\), with\(1\leq i\leq N\).
We provide some properties of the p-Laplacian operator as follows.
-
(A1)
If \(1< p<2\), \(xy>0\) and \(|x|,|y|\geq m>0\), then
$$ \bigl\vert \phi _{p}(x)-\phi _{p}(y) \bigr\vert \leq (p-1)m^{p-2} \vert x-y \vert ; $$ -
(A2)
If \(p>2\), \(xy>0\) and \(|x|,|y|\leq M\), then
$$ \bigl\vert \phi _{p}(x)-\phi _{p}(y) \bigr\vert \leq (p-1)M^{p-2} \vert x-y \vert . $$
Next, we find a solution of the linear variant of boundary value problem (1.1)–(1.2) as shown in the following lemma.
Lemma 2.3
For\(i,j\in \{1,2\}\)and\(i\neq j\), let\(\varLambda \neq 0\), \(\alpha _{i},\beta _{i},\theta _{i}\in (0, 1)\), \(\alpha _{i}+\beta _{i}\in (1,2]\), \(\lambda _{i}>0\)be given constants, \(h_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R} )\)and\(g_{i}\in C (\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R}^{+} )\)be given functions. Then the linear variant problem given by
has the unique solution\((u_{1},u_{2})\), where
where\(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\), the constantΛis defined by
and the functionals\({\mathcal{P}} [h_{1},h_{2}]\), \({\mathcal{Q}}[h_{1},h_{2}]\)are defined by
Proof
For \(i,j\in \{1,2\}\) and \(i\neq j\), taking the fractional sum of order \(\alpha _{i}\) for (2.1), we have
for \(t\in \mathbb{N}_{\alpha _{i}-1,T+\alpha _{i}}\).
From boundary condition (2.2), it implies that
Then from (2.9) we have
Next, taking the fractional sum of order \(\beta _{i}\) for (2.10), we have
for \(t\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\).
Using the fractional sum of order \(\theta _{i}\) for (2.11), we get
for \(t\in \mathbb{N}_{\alpha _{i}+\beta _{i}+\theta _{i}-3,T+\alpha _{i}+ \beta _{i}+\theta _{i}}\).
Using boundary condition (2.3) implies
and
\(C_{11}\), \(C_{12}\) can be represented by solving equations (2.13) and (2.14) as
and
where Λ, \({\mathcal{P}(h_{1},h_{2})}\) and \({\mathcal{Q}(h_{1},h_{2})}\) are defined as (2.6)–(2.8), respectively.
After substituting \(C_{11}\) and \(C_{12}\) into (2.11), we obtain (2.4) and (2.5). □
3 Existence and uniqueness result
In this section, we study the existence and uniqueness result for problem (1.1)–(1.2). For each \(i,j \in \{1,2\}\) and \(i\neq j\), we let \(E_{i}:C ( \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+ \beta _{i}}, \mathbb{R} )\) be the Banach space for all functions on \(\mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\). Clearly, the product space \(\mathcal{C}=E_{1}\times E_{2}\) is the Banach space. Define the spaces
with the norm
where
Obviously, the space \(( {\mathcal{C}_{1}\cap \mathcal{C}_{2}},\|(u_{1},u_{2})\|_{ \mathcal{C}_{1}\cap \mathcal{C}_{2}} )\) is also the Banach space with the norm
Let \({\mathcal{U}}=\mathcal{C}_{1}\cap \mathcal{C}_{2}\). The operator \(\mathcal{T}:{\mathcal{U}}\rightarrow {\mathcal{U}}\) is defined by
and
where \(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\), Λ is defined as (2.6), and the functionals \({\mathcal{P} [F_{1},F_{2}]}(u_{1},u_{2})\), \({\mathcal{Q}[F_{1},F_{2}]}(u_{1},u_{2})\) are defined by
with
For each \(i,j \in \{1,2\}\) and \(i\neq j\), we define the operators \((\mathcal{T}_{i}^{0}(u_{1},u_{2}))(t_{1},t_{2})\) and \((\mathcal{T}_{i}^{*}(u_{1},u_{2}))(t_{1}, t_{2})\) by
and
where \(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\), and the functionals \({\mathcal{P}^{*} [F_{1},F_{2}]}(u_{1},u_{2})\), \({\mathcal{Q}^{*}[F_{1},F_{2}]}(u_{1},u_{2})\) are defined by
Let \(\mathcal{T}_{i}=\mathcal{T}_{i}^{*} \circ \mathcal{T}_{i}^{0}\), then \(\mathcal{T}_{i}\) and \(\mathcal{T}:\mathcal{U}\rightarrow \mathcal{U}\) are continuous and compact operators. Note that problem (1.1)–(1.2) has solutions if and only if the operator \(\mathcal{T}\) has fixed points.
In the case \(p>2\), we have \(1< q<2\) due to \(\frac{1}{p}+\frac{1}{q}=1\) and the following theorem is obtained.
Theorem 3.1
Let\(p>2\)for each\(i,j\in \{1,2\}\), \(i\neq j\), \(F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}} \times\mathbb{R}^{3}, \mathbb{R} )\), \(\varphi _{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}},[0,\infty ) )\)with\(\varphi ^{o}_{i}=\max \{\varphi (t_{i}-1,s) \}\). In addition, suppose that:
-
(H1)
There exist constants\(\chi _{i}>0\)and\(0<\delta <\frac{1}{2-q}\)such that
$$\begin{aligned} \chi _{i}\Delta _{C}^{\alpha } \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta }\leq F_{i} [t_{i},t_{j},x,y,z ] \end{aligned}$$for any\(( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}\).
-
(H2)
There exist constants\(L_{i},M_{i},N_{i}>0\)such that, for each\(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\)and\(u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\in \mathbb{R}\),
$$\begin{aligned} &\bigl\vert F_{i} [t_{i},t_{j},u_{1},u_{2},u_{3} ]-F_{i} [t_{i},t_{j},v_{1},v_{2},v_{3} ] \bigr\vert \\ &\quad \leq L_{i} \vert u_{1}-v_{1} \vert +M_{j} \vert u_{2}-v_{2} \vert +N_{j} \vert u_{3}-v_{3} \vert . \end{aligned}$$ -
(H3)
\(g_{i}< g_{i}(t_{i})< G_{i}\)for each\(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+\alpha _{i}+\beta _{i}}\).
Then problem (1.1)–(1.2) has a unique solution provided that
where
Proof
For each \(i,j\in \{1,2\}\), \(i\neq j\), by (H1) we have
By (A1), (H2), and the definition of operator \(\mathcal{T}_{i}^{0}\), for any \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}\), we have
Using (3.18) and (H3), we have
and
From (3.19)–(3.20), it implies that
Similarly, we can find that
Next, taking the fractional difference of order \(\gamma _{1}\), \(\gamma _{2}\) for (3.2) and (3.3), respectively, we obtain
and
where \(t_{i}\in \mathbb{N}_{\alpha _{i}+\beta _{i}-\gamma _{i}+1,T+\alpha _{i}+ \beta _{i}-\gamma _{i}}\). Therefore,
Similarly, we obtain
From (3.22) and (3.25), we find that
In addition, by (3.21) and (3.26), we find that
Hence, from (3.27) and (3.28), we can conclude that
By (3.13), \(\mathcal{T}\) is a contraction mapping. Hence, by the Banach fixed point theorem, we get that \(\mathcal{T}\) has a fixed point, which is a unique solution of problem (1.1)–(1.2). □
In the same manner as Theorem 3.1, we can obtain the following theorem.
Theorem 3.2
Let\(p>2\), (H2)–(H3) hold, and the following condition hold:
-
(H4)
There exist constants\(\chi _{i}>0\)and\(0<\delta <\frac{1}{2-q}\)such that
$$ F_{i} [t_{i},t_{j},x,y,z ] \leq -\chi _{i}\Delta _{C}^{\alpha } \bigl( t_{i}^{\underline{\alpha _{i}}} \bigr)^{\delta} $$for any\(( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}\).
Then problem (1.1)–(1.2) has a unique solution.
In the case \(1< p<2\) and \(q>2\) since \(\frac{1}{p}+\frac{1}{q}=1\), we obtain the following theorem.
Theorem 3.3
Let\(1< p<2\)and (H2)–(H3) hold. For each\(i,j\in \{1,2\}\), \(i\neq j\), \(F_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}\times \mathbb{R}^{3}, \mathbb{R} )\), \(\varphi _{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}, [0,\infty ) )\)with\(\varphi ^{o}_{i}=\max \{\varphi (t_{i}-1,s) \}\). Suppose that the following assumption holds:
-
(H5)
There exists a nonnegative function\(k_{i}\in C (\mathbb{N}_{\alpha _{1}+\beta _{1}-2,T+\alpha _{1}+ \beta _{1}}\times \mathbb{N}_{\alpha _{2}+\beta _{2}-2,T+\alpha _{2}+ \beta _{2}}, [0,\infty ) )\)and\(\mathcal{M}_{i}:=\frac{1}{\varGamma (\alpha _{i})} \sum_{ \xi =\alpha _{i}+\beta _{i}-1}^{T+\alpha _{i}+\beta _{i}-1}(T+2 \alpha _{i}+\beta _{i}-1-\sigma (\xi ))^{\underline{\alpha _{i}-1}} k_{i}(T+ \alpha _{j}+\beta _{j},\xi )>0\)such that
$$ F_{i}[ t_{i},t_{j},x,y,z]\leq k_{i}(t_{i},t_{j}) $$for any\(( t_{i},t_{j},x,y,z )\in \mathbb{N}_{\alpha _{i}+\beta _{i}-2,T+ \alpha _{i}+\beta _{i}}\times \mathbb{N}_{\alpha _{j}+\beta _{j}-2,T+ \alpha _{j}+\beta _{j}}\times \mathbb{R}^{3}\).
Then problem (1.1)–(1.2) has a unique solution provided that
where\(\mathcal{K}_{i}\)is defined as (3.14), and
Proof
For each \(i,j\in \{1,2\}\), \(i\neq j\), by (H5) we have
By (A2), (H2), and the definition of operator \(\mathcal{T}_{i}^{0}\), for any \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}\), we have
Then, by (3.19) and (3.20), we have
Similarly as in Theorem 3.1, we obtain
Therefore, by (3.39) and (3.40), we can conclude that
By (3.30), we can conclude that \(\mathcal{T}\) is a contraction mapping. Hence, by the Banach fixed point theorem, \(\mathcal{T}\) has a fixed point, which is a unique solution of problem (1.1)–(1.2). □
4 Some examples
In this section, we consider some examples to illustrate our main result.
Example 4.1
Consider the following fractional sum boundary value problem:
subject to nonlocal fractional sum boundary conditions of the form
where \(t\in {\mathbb{N}}_{0,10}\). Functions \(F_{1}\), \(F_{2}\) are determined by
and
Here, \(p=\frac{5}{2}\), \(q=\frac{5}{3}\), \(\alpha _{1}=\frac{1}{2}\), \(\alpha _{2}= \frac{3}{4}\), \(\beta _{1}=\frac{2}{3}\), \(\beta _{2}=\frac{3}{8}\), \(\gamma _{1}= \frac{1}{3}\), \(\gamma _{2}=\frac{2}{3}\), \(\omega _{1}=\frac{3}{4}\), \(\omega _{2}=\frac{1}{4}\), \(\theta _{1}=\frac{1}{4}\), \(\theta _{2}= \frac{2}{5}\), \(\eta _{1}=\frac{19}{6}\), \(\eta _{2}=\frac{41}{8}\), \(\lambda _{1}=2\), \(\lambda _{2}=3\), \(T=10\), \(g_{1}(t_{1})=e^{\cos t_{1}\pi }\), \(g_{2}(t_{2})=e^{2 \sin t_{2}\pi }\), \(\varphi _{1}(t_{1},s)= \frac{e^{-s}}{(t_{1}+10)^{3}}\), \(\varphi _{2}=\frac{e^{-s}}{(t_{2}+20)^{2}}\), and \(\varphi _{1}^{o}=\frac{216}{166\text{,}375 e^{1/6} }\approx 0.0011\), \(\varphi _{2}^{o}= \frac{64}{23\text{,}409 e^{1/8} }\approx 0.0024\).
Let \(t_{1}\in {\mathbb{N}}_{-\frac{5}{6},\frac{67}{6}}\) and \(t_{2}\in {\mathbb{N}}_{-\frac{7}{8},\frac{89}{8}}\). Taking \(\chi _{1}=3\), \(\chi _{2}=2\) and \(1=\delta <\frac{1}{2-q}=3\), we have
Thus, (H1) holds.
For \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}\), we have
Thus, (H2) holds with \(L_{1}=6.211\times 10^{-6}\), \(L_{2}=9.307\times 10^{-6}\), \(M_{1}=4.141 \times 10^{-6}\), \(M_{2}=1.889\times 10^{-6}\), \(N_{1}=2.856\times 10^{-6}\), and \(N_{2}=4.653\times 10^{-6}\).
Since \(\frac{1}{e}\leq g_{1}(t_{1})\leq e \) and \(\frac{1}{e^{2}}\leq g_{2}(t_{2})\leq e^{2}\).
Thus, (H3) holds with \(g_{1}=\frac{1}{e}\), \(g_{2}=\frac{1}{e^{2}}\) and \(G_{1}=e\), \(G_{2}=e^{2}\).
Finally, we find that
Therefore, we have
Hence, by Theorem 3.1, boundary value problem (4.1)–(4.2) has a unique solution.
Example 4.2
Consider the following fractional sum boundary value problem:
where \(t\in {\mathbb{N}}_{0,10}\), and the nonlocal fractional sum boundary conditions satisfy (4.2). Functions \(H_{1}\), \(H_{2}\) are determined by
where \(\varPsi ^{\frac{3}{4}} u_{1}\), \(\varPsi ^{\frac{1}{4}} u_{2}\) are defined as (4.3) and (4.4), respectively.
Let \(t_{1}\in {\mathbb{N}}_{-\frac{5}{6},\frac{67}{6}}\) and \(t_{2}\in {\mathbb{N}}_{-\frac{7}{8},\frac{89}{8}}\). Using \(g_{1}(t_{1},t_{2})=\frac{3t_{1}^{\underline{2}}}{500\text{,}000e^{7}}+ \frac{2t_{2}^{\underline{2}} }{400\text{,}000e^{8}}\) and \(g_{2}(t_{1},t_{2})=\frac{2t_{2}^{\underline{2}}}{6\text{,}000\text{,}000e^{6}}+ \frac{t_{1}^{\underline{2}} }{5\text{,}000\text{,}000e^{7}}\), we have
For \((u_{1},u_{2}),(v_{1},v_{2})\in \mathcal{C}\), we have
Thus, (H2) holds with \(L_{1}=6.211\times 10^{-7}\), \(L_{2}=9.307\times 10^{-7}\), \(M_{1}=N_{1}=2.070 \times 10^{-7}\), and \(M_{2}=N_{2}=9.447\times 10^{-8}\).
From Example 4.1, we get \(\varLambda \geq 0.029\), \(g_{1}=\frac{1}{e}\), \(g_{2}=\frac{1}{e^{2}}\) and \(G_{1}=e\), \(G_{2}=e^{2}\).
Finally, we find that
Hence,
From Theorem 3.3, we can conclude that boundary value problem (4.5) and (4.2) has a unique solution.
5 Conclusions
We have proved existence and uniqueness results of the nonlocal fractional sum boundary value problem for a coupled system of fractional sum-difference equations with p-Laplacian operator (1.1)–(1.2) by using the Banach fixed point theorem. Our problem contains both Riemann–Liouville and Caputo fractional difference with five fractional differences and four fractional sums.
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The first author of this research was supported by Kasetsart University. Furthermore, the last author of this research was supported by Suan Dusit University.
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This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-028.
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Siricharuanun, P., Chasreechai, S. & Sitthiwirattham, T. On a coupled system of fractional sum-difference equations with p-Laplacian operator. Adv Differ Equ 2020, 361 (2020). https://doi.org/10.1186/s13662-020-02826-3
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DOI: https://doi.org/10.1186/s13662-020-02826-3