 Research
 Open Access
 Published:
Existence and uniqueness of solutions for a mixed pLaplace boundary value problem involving fractional derivatives
Advances in Difference Equations volume 2020, Article number: 694 (2020)
Abstract
In this article, the existence and uniqueness of solutions for a multipoint fractional boundary value problem involving two different left and right fractional derivatives with pLaplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.
1 Introduction
In recent years, fractionalorder calculus theory has been widely used in mathematics, science, engineering, etc. As a result, the studies of such equation have gained considerable popularity, see [1–8]. Also, fractionalorder mixed differential or integral equation involving different fractional derivatives, such as conformable fractional, Riemann–Liouville, and Caputo, has got a lot of interest, and even fractionalorder differential or integral equations with pLaplace operator have been extensively discussed by more and more researchers [9–14].
In [11], a mixed fractional pLaplace boundary value problem was studied by Liu et al.
where \(\varphi _{p}(t)= \vert t \vert ^{p2}\cdot t\), \(p>1\), \(1< \alpha , \beta \leq 2\), \(r_{1}, r_{2} \geq 0\), \(D_{0+}^{\alpha }\) is Riemann–Liouville fractional derivative, \({}^{c} D_{0+}^{\beta }\) is Caputo fractional derivative, and \(f:[0,1] \times [0, +\infty ) \times (\infty , 0]\rightarrow [0, + \infty )\) is continuous. The minimum upper solution and the maximum lower solution of the above boundary value problem were given by applying the lower and upper solutions method.
In [2], Bai investigated the uniqueness and existence of solutions of the following fractionalorder differential equation:
where \(D_{0+}^{\alpha }\) is Riemann–Liouville fractional derivative, \({}^{c} D_{0+}^{\beta }\) is Caputo fractional derivative, \(0< \beta \leq 1\), \(2 < \alpha \leq 2+ \beta \), \(p> 1\), and \(f:[a,b] \times \mathbb{R} \rightarrow \mathbb{R}\) is continuous. The uniqueness result of a solution and the existence of specific solutions to problem were showed by applying Guo–Krasnoselskii’s fixed point theorem.
In [4], Dang et al. proposed a fresh approach to gain the existence and uniqueness of solutions of a fourthorder twopoint nonlinear differential equation
where \(f:[0,1] \times \mathbb{R}^{2} \rightarrow \mathbb{R}\) is continuous. In [3], Bai et al. considered a class of fourthorder nonlinear differential equation with pLaplace operator, and the boundary conditions change from two points to multiple points compared to the above problem. In both papers, the main results were given by applying the Banach contraction mapping principle.
As far as we know, nobody used the method which was put forward by Dang [6] and Bai [3] to prove the existence and uniqueness of solutions of a nonlinear multipoint fractionalorder pLaplace boundary value problem, which has at least two different kinds of fractional derivatives. Inspired by the abovementioned articles, the following mixed boundary value problem is studied in this work:
where \(1< \gamma , \delta \leq 2\), \(0<\mu ,\eta < 1\), \(0\leq r_{1} < \frac{1}{\mu ^{\delta  1}}\), \(0\leq r_{2}< \frac{1}{1 \eta }\), \(\varphi _{p}(x)= \vert x \vert ^{p2}\cdot x\), \(\frac{1}{p}+ \frac{1}{q}= 1\), \(p, q> 1\), \(D_{0+}^{\delta }\) is left Riemann–Liouville fractional derivative, \({}^{c} D_{1}^{\gamma }\) is right Caputo fractional derivative, and function \(g\in C([0,1] \times \mathbb{R}^{2}, \mathbb{R})\).
The rest of this work is organized as follows. In Sect. 2, some related definitions and necessary lemmas of fractional calculus theory are presented, which will be applied in the main results of this article. In Sect. 3, the existence and uniqueness of solutions of the mixed fractionalorder pLaplace differential equation are proved by applying the method which was put forward by Dang and Bai. In Sect. 4, a particular example is constructed to verify the main conclusions of the paper.
2 Preliminaries
This section introduces some related definitions and necessary lemmas of fractional calculus theory.
Definition 2.1
For given \(\gamma > 0\) and the function \(y:(0,\infty )\rightarrow \mathbb{R}\), define the left and right Riemann–Liouville fractional integrals respectively as follows:
Definition 2.2
For given \(\gamma > 0\), \(\gamma \in (n, n+ 1)\) and the function \(y:(0,\infty )\rightarrow \mathbb{R}\), define the left Riemann–Liouville fractional derivative and the right Caputo fractional derivative respectively as follows:
Let \(E= C[0,1]\), whose norm \(\Vert \cdot \Vert \) is the maximum norm. Given \(\phi \in C[0,1]\) and the constants \(r_{1},r_{2}\in \mathbb{R}\), discuss the following fractionalorder mixed pLaplace boundary value problem:
where \(1< \gamma , \delta \leq 2\), \(0<\mu ,\eta < 1\), \(0\leq r_{1}< \frac{1}{\mu ^{\delta  1}}\), \(0\leq r_{2} < \frac{1}{1 \eta }\).
Lemma 2.1
The unique solution of the fractionalorder mixed pLaplace boundary value problem (2.1) is equivalent to
where
and
with
Moreover, \(G_{1}(x, \tau )>0\), \(G_{2}(\tau , s)>0\) for \(x,\tau ,s\in (0,1)\).
Proof
Let \(\varphi _{p} (D_{0+}^{\delta } y(x))= k(x)\), then the mixed boundary value problem (2.1) changes to
and
Reduce the equation \({}^{c} D_{1}^{\gamma } k(x)= \phi (x)\) as an equivalent equation
Using the condition \(k(1)=0\) yields \(c_{1}= 0\). Since \(k(0)= r_{2}k(\eta )\), then
and
By calculation, we can get
So,
Similarly, the solution of boundary value problem (2.4) is given by
Consequently, boundary value problem (2.1) is equivalent to (2.2).
From the monotonicity, for \(x,\tau ,s\in (0,1)\), \(G_{1}(x, \tau ),G_{2}(\tau , s)>0\) are verified easily. The proof is completed. □
Lemma 2.2
For given \(\phi (s) \in C[0,1]\), set
Then
Proof
Since \(\frac{1}{p}+ \frac{1}{q}= 1\) and \(\varphi _{p}\) is increasing, then
So, \(\Vert w \Vert \leq M_{2}^{q1} \Vert \phi \Vert ^{q1}\). Similarly, \(\Vert y \Vert \leq M_{1}M_{2}^{q1} \Vert \phi \Vert ^{q1}\). The proof is completed. □
Lemma 2.3
The following relations of pLaplace operator hold:

(i)
There are \(1< p\leq 2\), \(\vert k_{1} \vert , \vert k_{2} \vert \geq n> 0\), and \(k_{1}k_{2}> 0\) such that
$$\begin{aligned}& \bigl\vert \varphi _{p}(k_{2}) \varphi _{p}(k_{1}) \bigr\vert \leq (p 1)n^{p2} \vert k_{2}k_{1} \vert ; \end{aligned}$$ 
(ii)
There are \(p> 2\) and \(\vert k_{1} \vert , \vert k_{2} \vert \leq N\) such that
$$\begin{aligned}& \bigl\vert \varphi _{p}(k_{2}) \varphi _{p}(k_{1}) \bigr\vert \leq (p 1)N^{p2} \vert k_{2}k_{1} \vert . \end{aligned}$$
3 Main results
This section studies the existence and uniqueness of solutions for mixed fractionalorder pLaplace boundary value problem (1.1) by applying the Banach contraction mapping principle.
Given number \(M> 0\), denote
and by a closed ball \(B[O,M]\) in the space of continuous functions \(C[0,1]\).
Theorem 3.1
Assume that \(1< p\leq 2\), and there exist some numbers \(M, Q_{1}, Q_{2}>0\) such that the following conditions hold:

(H1)
\(\vert g(x, y, w) \vert \leq M\) for \((x, y, w)\in D_{M}\);

(H2)
\(\vert g(x,y_{2},w_{2})  g(x,y_{1},w_{1}) \vert \leq Q_{1} \vert y_{2}y_{1} \vert + Q_{2} \vert w_{2}w_{1} \vert \) for \((x,y_{i},w_{i})\in D_{M}\), \(i=1,2\);

(H3)
\(L_{1}:= (q1) M^{q2} M_{2}^{q1} (Q_{1} M_{1} + Q_{2})<1\).
Then the mixed boundary value problem (1.1) has a unique solution satisfying the estimation
Proof
First of all, define an operator \(A:C[0,1]\rightarrow C[0,1]\) by
From the continuity of \(G_{1}(x,\tau )\), \(G_{2}(\tau ,s)\), and \(g(x,y,w)\), it is not hard to see that the operator A is a continuous operator. According to Lemma 2.1, it is easy to get the following conclusion that if the mixed boundary value problem (1.1) has a solution \(y(x)\), then \(\phi (x)= {}^{c} D_{1}^{\gamma }( \varphi _{p} (D_{0+}^{\delta } y(x)))\) is the fixed point of the operator A. On the contrary, if \(\phi (x)\) is a fixed point of the operator A, then
also is the solution of the mixed boundary value problem (1.1).
Next, what we need to prove is that the operator A maps \(B[O,M]\) into itself. Given \(\phi (x)\in B[O,M]\), by Lemma 2.2, there is
Consequently, for any \(x\in [0,1]\), there is \((x,y(x),w(x))\in D_{M}\). So, from \((H1)\), we can conclude that
therefore \((A\phi )(x)\in B[O, M]\). Namely, the operator A is an operator that maps \(B[O,M]\) into itself.
Finally, the operator \(A:B[O,M]\rightarrow B[O,M]\) is proven to be a contraction mapping. Obviously, \(B[O,M]\) is proven to be a complete distance space on account of that \(B[O,M]\) is a subspace of \(C([0,1], \Vert \cdot \Vert )\). From \((H2)\), Lemma 2.2, and (ii) of Lemma 2.3, there is \(\vert \int _{0}^{1} G_{2}(\tau ,s) \phi (s) \,ds \vert \leq M_{2}M := N\) for each \(\phi _{1}(x), \phi _{2}(x)\in B[O,M]\) and \(1< p\leq 2\), that is, \(q \geq 2\), we gain
Hence,
where \(L_{1}<1\) is given by \((H3)\). Therefore, the operator \(A:B[O,M]\rightarrow B[O,M]\) is a contraction mapping. That is to say, there is a unique solution y(x) of the mixed boundary value problem (1.1) such that
The proof is completed. □
4 An example
Example 4.1
Consider the following mixed fractionalorder pLaplace differential equation:
where \(g(x,y,w)= 2y^{2}w + 3y  2w + 4\sin (\pi x)\).
We choose \(p=2\), that is, \(q=2\). By a simple computation, we obtain that \(M_{1}=\frac{1}{8}\), \(M_{2}=\frac{1}{3}\). Then a suitable number \(M > 0\) is chosen to satisfy all the conditions of Theorem 3.1, and
Obviously, for each \((x,y,w)\in D_{M}\), there is
We can easily get that \(0 < M < 13.4164\). Therefore, choose \(M = 1\), condition \((H1)\) holds.
Meanwhile, for \((x,y,w)\in D_{M}\),
So we choose \(Q_{1}=3.5\), \(Q_{2}=2.5\), condition \((H2)\) holds.
Moreover,
condition \((H3)\) holds.
So, the above boundary value problem has a unique solution satisfying
5 Conclusion
In this article, we have discussed a novel approach which has been put forward by Dang and Bai to prove the existence and uniqueness of solution for a nonlinear multipoint fractionalorder pLaplace differential equation, which has at least two different kinds of left and right fractional derivatives. As far as we know, almost nobody explored the work in this area. The advantage of this method is that it is very easy to verify and can be applied to many conditions. Of course, if the boundary condition is changed, the assumptions can be weakened appropriately. Last but not least, the approach mentioned above can be applied to some other boundary value problems, such as conformable fractional order. Furthermore, the boundary conditions can also become integral boundary conditions, and so on.
Availability of data and materials
Data sharing not applicable to this article as no data sets were generated or analysed during the current study.
References
Ahmad, B., Broom, A., Alsaedi, A., Ntouyas, S.K.: Nonlinear integrodifferential equations involving mixed right and left fractional derivatives and integrals with nonlocal boundary data. Mathematics 8(3), 336 (2020)
Bai, C.Z.: Existence and uniqueness of solutions for fractional boundary value problems with pLaplacian operator. Adv. Differ. Equ. 2018, 4 (2018)
Bai, Z.B., Du, Z.J., Zhang, S.: Iterative method for a class of fourthorder pLaplacian beam equation. J. Appl. Anal. Comput. 9(4), 1–11 (2019)
Dang, Q.A., Dang, Q.L., Quy, N.: A novel efficient method for nonlinear boundary value problems. Numer. Algorithms 76, 427–439 (2017)
Dang, Q.A., Nguyen, T.H.: Solving the Dirichlet problem for fully fourth order nonlinear differential equation. Afr. Math. 30, 623–641 (2019)
Dang, Q.A., Quy, N.: New fixed point approach for a fully nonlinear fourth order boundary value problem. Bol. Soc. Parana. Mat. 36, 209–223 (2018)
Filippucci, R., Pucci, P., Radulescu, V.: Existence and nonexistence results for quasilinear elliptic exterior problems with nonlinear boundary conditions. Commun. Partial Differ. Equ. 33, 706–717 (2008)
Lakoud, A.G., Khaldi, R., Kılıçman, A.: Existence of solutions for a mixed fractional boundary value problem. Adv. Differ. Equ. 2017, 164 (2017)
Liang, S.H., Zhang, J.H.: Positive solutions of boundary value problems of nonlinear fractional differential equation. Nonlinear Anal., Model. Control 71, 5545–5550 (2009)
Liu, X.P., Jia, M.: Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives. Appl. Math. Comput. 353, 230–242 (2019)
Liu, X.P., Jia, M., Ge, W.G.: The method of lower and upper solutions for mixed fractional fourpoint boundary value problem with pLaplacian operator. Appl. Math. Lett. 65, 56–62 (2017)
Liu, X.P., Jia, M., Xiang, X.F.: On the solvability of a fractional differential equation model involving the pLaplacian operator. Comput. Math. Appl. 64, 3267–3275 (2012)
Reed, M., Simon, B.: IV. Analysis of Operators. Methods of Modern Mathematical Physics. Academic Press, New York (1978)
Xu, M.R., Sun, S.R.: Positivity for integral boundary value problems of factional differential equations with two nonlinear terms. J. Appl. Anal. Comput. 59, 271–283 (2019)
Acknowledgements
Not applicable.
Funding
This work is supported by NSFC (11571207), Taishan Scholar project, and SDUST graduate innovation project SDKDYC170343.
Author information
Authors and Affiliations
Contributions
All authors read and approved the final manuscript. All authors contributed equally to the writing of this paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
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/.
About this article
Cite this article
Wang, S., Bai, Z. Existence and uniqueness of solutions for a mixed pLaplace boundary value problem involving fractional derivatives. Adv Differ Equ 2020, 694 (2020). https://doi.org/10.1186/s13662020031542
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662020031542
MSC
 34A08
 34B15
 35J05
Keywords
 Fractional derivatives
 pLaplace operator
 Existence and uniqueness