Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 3543670, 12 pages
https://doi.org/10.1155/2019/3543670

Research Article

Positive Solutions to a Coupled Fractional Differential System with -Laplacian Operator

Yuehan Liu, Xiaodi Zhao, and Huihui Pang

College of Science, China Agricultural University, Beijing 100083, China

Correspondence should be addressed to Huihui Pang; phh2000@163.com

Received 4 April 2019; Accepted 26 May 2019; Published 17 June 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 Yuehan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we consider a fractional differential system with multistrip and multipoint mixed boundary conditions involving -Laplacian operator and fractional derivatives. The existence result of positive solutions is established by the Leggett-Williams fixed point theorem. Also, an example is presented to illustrate our main result.

1. Introduction

Fractional calculus is a significant branch of mathematical analysis which concerns derivatives and integrals. Fractional differential equations are caused by the intensive development of the theory of fractional calculus and have gained great interest currently, because of their crucial applications in various fields of physics, biology, and engineering such as control, porous media, electromagnetic, and other fields [1–4].

Moreover, many mathematical formulations of physical and biological phenomena lead to fractional differential equations with operator, such as memory, fluid dynamics, predator harvesting, and hereditary properties of various materials, which are difficult to achieve in classical integer-order models. Hence, in recent years, more and more researchers have considered the fractional differential equations with -Laplacian operator, by applying various mathematical theory and tools; refer to [5–9].

For example, authors in [10] discussed the following differential system with -Laplacian operator

where , , , , , is a operator: , and . Authors established the conclusion of the existence and uniqueness of solution to the above problem by using topological degree theory.

By reading the existing literatures [11–24], we note that the fractional differential system involving -Laplacian operator and multistrip and multipoint boundary conditions has not been studied yet. Thus, in this paper, we first pay close attention to the following fractional differential system, involving -Laplacian operator and lower fractional derivatives:

with multistrip and multipoint boundary conditions:

where and are the Riemann-Liouville fractional derivatives of order and , is a continuous function. For , , , , , , , , are nonnegative constants. . And , are nondecreasing functions of bounded variation in .

We emphasize that, despite the complication of the discrete and integral boundary conditions (3) and (4), operator T can be presented in a concise form based on the piecewise form of Green’s function. And we can accurately estimate the upper and lower bounds of its value, which is fully prepared for the establishment of the main theorem.

Accordingly, the conclusion we reached is an extensive result and a meaningful supplement to the theory of the fractional differential systems with -Laplacian operator.

Theorem 1 (the Leggett-Williams fixed point theorem). Let be a cone in a real Banach space , , be a nonnegative continuous concave function on P such that for all , and . Suppose that is completely continuous and there exist constants such that

 ; ; 

Then has at least three fixed points , , and which satisfy

2. Preliminaries

In this section, we will put forward some crucial definitions and theorems, preparing for establishing the existence of solutions.

Definition 2. The Riemann-Liouville fractional integral of order of a function is given by

provided the right-hand side is pointwise defined on , where is the Euler gamma function defined by , for

Definition 3. The Riemann-Liouville fractional derivative of order for a function is given by ,

where , stands for the largest integer not greater than .

According to the definitions of Riemann-Liouville’s derivative, the following lemmas can be achieved.

Lemma 4. For , if we assume that , then we have

for some , while is the smallest integer greater than or equal to .

Remark 5. The following properties are useful for our discussion:

(i)As a basic example, we quote for ,

(ii).

For convenience, we denote

Besides, it should be noted that .

Lemma 6. For , the equations

with boundary conditions (3) and (4) have a unique solution as follows:

where

and is a piecewise function; the expression of the th piece is as follows

Meanwhile, is also a piecewise function; the expression of th piece is as follows

Proof. We shall reduce BVP (11), (3), and (4) to two fractional differential problems. To this goal, first, by means of transformation, we convert problem (11) into

In view of Lemma 4, the solution of problem (17) can be written as

where , are arbitrary real constants.
By , we find . Setting in (19), together with the boundary condition , we have

Accordingly, we can easily find out

Combining (21) with (19), we can obtain

Next, considering the multipoint boundary conditions, we divide the domain of integration into pieces. For , the unique solution of (17) can be expressed as

When , we observe that the th integrand in (23) can be taken as follows:

where is also noted by 0.
As a result, the Green function is always valid and can exactly express each integrand in the above formula, for . Thus, we can take (23) as a unified form

is a piecewise function such that the form of the th piece is given by (15).
We take (25) into (18); then

From Lemma 4, we have

where , are arbitrary real constants.
By , we easily find . Setting t=1 in (27), together with the boundary condition , we have

Set and into (27); we find out

Furthermore, by integrating and summing on both sides of (29), it is easy to see that

Then, we get

Take (31) into (29); we conclude that the unique solution of BVP (11), (3), and (4) is

Analogically, the unique solution of BVP (11), (3), and (4) is

where is also a piecewise function, the th piece is defined by (16), and the calculation process is similar.
This completes the proof.

Denote

Lemma 7. For and , it holds that

Proof. For and , we have , while is a monotone increasing function. Therefore, we can infer that . For , we finally find out , which implies and , for , .
This completes the proof.

Lemma 8. For , the functions defined by (14) have the following properties:

(1), , for ;(2); 

Proof. (1) The positive of and is easily checked.
(2) For , we have

For , we have

The other inequality is achieved similarly. All these complete the proof of the lemma.

For , denote

and the constant is given by the maximum value of , , , and .

Lemma 9. For , the functions , , and defined by Lemma 6 satisfy the following results:

(1), , are continuous, , , ;(2), ;(3), .

Proof. The nonnegativeness of , , and is easily checked.
(2) According to the property (2) in Lemma 8, we have

For , we have

Because of the randomicity of , we have

According to the above, from the property (2) of Lemma 8, we easily get

This completes the proof of the lemma.

Define the space and is endowed with the norm , where ,

Then we introduce the product space endowed with the norm and define a partial order where , , for .

Lemma 10. is a Banach space.

Proof. Let be a Cauchy sequence in the space . Then clearly and are Cauchy sequences in the space . Therefore, and converge to some and on uniformly and . Then, we need to prove .
Note that

By the convergence of , we get

uniformly for . On the other hand, by Remark 5, one has , for and some . Furthermore, we can get

From (44) and (45),we obtain

Taking the -order derivative on both sides of yields, we have

In view of Remark 5 and Lemma 4, we find that

Above all, we can achieve that is a Banach space. This completes the proof.

Define the cone by

Let the nonnegative continuous concave functional on the cone be defined by

where is a positive constant, satisfying .

For convenience, for , we denote

where .

For any , we introduce the integral operator by

where

Consequently, by calculation

where

Lemma 11. is completely continuous.

Proof. Firstly, we prove that . In view of the nonnegativeness and continuity of , and , is continuous and for . Hence, .
Next, we show is uniformly bounded. Let be bounded; that is, there exists a positive constant such that , , , for all . Let . Then for , from the Lemma 9, we have

Hence, is uniformly bounded. Then, we show is equicontinuous. Indeed, for any , , , we have

Moreover,

Therefore, the above process implies that is equicontinuous for all .
By means of the Arzela-Ascoli theorem, is completely continuous. Likewise, the proof of follows the same method as . Thus, is completely continuous.

3. Existence Result

For convenience, we denote

Theorem 12. There exist nonnegative numbers , and satisfies the following conditions:

, for ;, for ;, for

Then BVP (2)-(4) has at least three positive solutions , , and such that

Proof. We will verify that conditions of Theorem 1 are satisfied.
Let We first prove that . For any , , we have , , , . Consequently, combining with properties (2) and (3) of Lemma 9,

Thus, and further to get . Therefore is completely continuous. The proof of is the same as . Thus, is completely continuous.
By the above, condition of Theorem 1 can be satisfied, using assumption .
Finally, in order to verify , we set , for . It is easy to find that

Hence,

If , we have , , , , for all . Then

That is, for all . This shows that condition of Theorem 1 is also satisfied.
From the above, BVP (2)-(4) has at least three positive solutions , , and such that

The proof is completed.

Example 13. To verify the feasibility of our conclusion, here we consider the boundary value problem of nonlinear fractional differential system as follows:

with boundary conditions

In this model, we set , and

Let , for ,

It is obvious that () is continuous. Take ; by calculation, we obtain

For , set , , and , which satisfy , such that

, for ;, for ;, for ,

Thus, all the conditions in Theorem 12 are achieved. Consequently, BVP (66) has at least three positive solutions , , and such that , , and with .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the manuscript, read and approved the final draft.

Acknowledgments

The work is supported by National Training Program of Innovation (No. 201910019172). The funding body plays an important role in the design of the study and analysis, calculation, and writing the manuscript.

References

Y. Wei and Z. Bai, “Solvability of some fractional boundary value problems with a convection term,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 1230502, 6 pages, 2019.

L. Diening, P. Lindqvist, and B. Kawohl, “Mini-workshop: the p-laplacian operator and applications,” Oberwolfach Reports, vol. 10, no. 1, pp. 433–482, 2013.

C. Azizieh, P. Cle'ment, and E. Mitidieri, “Existence and a priori estimates for positive solutions of p-laplace systems,” Journal of Differential Equations, vol. 184, no. 2, pp. 422–442, 2002.

C. Azizieh and P. Cle'ment, “A priori estimates and continuation methods for positive solutions of p-laplace equations,” Journal of Differential Equations, vol. 179, no. 1, pp. 213–245, 2002.

X. Liu, M. Jia, and X. Xiang, “On the solvability of a fractional differential equation model involving the p-Laplacian operator,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3267–3275, 2012.

X. Liu, M. Jia, and W. Ge, “The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator,” Applied Mathematics Letters, vol. 65, pp. 56–62, 2017.

M. Feng, B. Du, and W. Ge, “Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3119–3126, 2009.

Y. Tian, S. Sun, and Z. Bai, “Positive solutions of fractional differential equations with p-Laplacian,” Abstract & Applied Analysis, vol. 2017, Article ID 3187492, 8 pages, 2017.

K. Sheng, W. Zhang, and Z. Bai, “Positive solutions to fractional boundary-value problems with p-laplacian on time scales,” Boundary Value Problems, vol. 2018, article 70, 2018.

T. Lv, H. Pang, and L. Cao, “Existence results for fractional differential equations with multistrip Riemann-Stieltjes integral boundary conditions,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 2352789, 8 pages, 2018.

J. Henderson and R. Luca, “Systems of Riemann-Liouville fractional equations with multi-point boundary conditions,” Applied Mathematics and Computation, vol. 309, pp. 303–323, 2017.

D. Gao and J. Li, “Existence results for impulsive fractional differential inclusions with two different Caputo fractional derivatives,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 1323176, 9 pages, 2019.

E. Cetin and F. S. Topal, “Existence of solutions for fractional four point boundary value problems with p-Laplacian operator,” Journal of Computational Analysis and Applications, vol. 19, no. 5, pp. 892–903, 2015.

Q. Li, C. Hou, L. Sun, and Z. Han, “Existence and uniqueness for a class of multi-term fractional differential equations,” Applied Mathematics and Computation, vol. 53, no. 1-2, pp. 383–395, 2017.

K. Shah and R. A. Khan, “Existence and uniqueness results to a coupled system of fractional order boundary value problems by topological degree theory,” Numerical Functional Analysis and Optimization, vol. 37, no. 7, pp. 887–899, 2016.

B. Di and H. Pang, “Existence results for the fractional differential equations with multi-strip integral boundary conditions,” Applied Mathematics and Computation, vol. 59, no. 1-2, pp. 1–19, 2019.

C. Gao, Z. Gao, and H. Pang, “Existence criteria of solutions for a fractional nonlocal boundary value problem and degeneration to corresponding integer-order case,” Advances in Difference Equations, vol. 2018, article 408, 2018.

Y. Sun and M. Zhao, “Positive solutions for a class of fractional differential equations with integral boundary conditions,” Applied Mathematics Letters, vol. 34, pp. 17–21, 2014.

M. Cui, Y. Zhu, and H. Pang, “Existence and uniqueness results for a coupled fractional order systems with the multi-strip and multi-point mixed boundary conditions,” Advances in Difference Equations, vol. 2017, article 224, 2017.

M. Feng, P. Li, and S. Sun, “Symmetric positive solutions for fourth-order n-dimensional m-Laplace systems,” Boundary Value Problems, vol. 2018, article 63, 2018.

X. Zhang, “Exact interval of parameter and two infinite families of positive solutions for a nth order impulsive singular equation,” Journal of Computational and Applied Mathematics, vol. 330, pp. 896–908, 2018.

M. Feng, “New results of coupled system of k Hessian equations,” Applied Mathematics Letters, vol. 94, pp. 196–203, 2019.

X. Liu and M. Jia, “Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives,” Applied Mathematics and Computation, vol. 353, pp. 230–242, 2019.

X. Liu and M. Jia, “The method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian,” Advances in Difference Equations, vol. 2018, article 28, 2018.