Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 2327580, 6 pages
https://doi.org/10.1155/2020/2327580

Research Article

Multiple Positive Solutions for Fractional Three-Point Boundary Value Problem with p-Laplacian Operator

Dong Li,¹ Yang Liu,² and Chunli Wang³

¹Department of Mathematics, College of Science Jiamusi University, Jiamusi, Heilongjiang 154007, China
²School of Mathematics and Statistics, Hefei Normal University, Hefei, Anhui 230061, China
³Institute of Information Technology of GUET, Guilin, Guangxi 540004, China

Correspondence should be addressed to Chunli Wang; wangchunliwcl821222@sina.com

Received 23 May 2020; Accepted 20 June 2020; Published 9 July 2020

Academic Editor: Jia-Bao Liu

Copyright © 2020 Dong Li 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 investigate the existence of multiple positive solutions or at least one positive solution for fractional three-point boundary value problem with p-Laplacian operator. Our approach relies on the fixed point theorem on cones. The results obtained in this paper essentially improve and generalize some well-known results.

1. Introduction

Nowadays, fractional calculus has been adapted to numerous fields, such as engineering, mechanics, physics, chemistry, and biology. Many essays and mongraphs studying various issues in fractional calculus have been researched (see [1–6]). In particular, fractional differential equations have been found to be a powerful tool in modeling various phenomena in many areas of science and engineering such as physics, fluid mechanics, and heat conduction. More details about research achievement on fractional differential equations and their applications are shown in [7–10].

Recently, fractional differential equations have gained considerable attention (see [11–15] and the references therein). Fractional differential equations and differential equations with p-Laplacian operators have attracted much attention from many mathematicians. As a result, meaningful research results have been drawn [16–20]. Fractional-order boundary value problems involving classical, multipoint, high-order, and integral boundary conditions have extensively been studied by many researchers and a variety of results can be found in recent literature on the topic [21–25].

In [15], Chai studied the boundary value problems of fractional differential equations with p-Laplacian operator as follows:

where , , , , and , , are the standard Riemann–Liouville derivatives. Some existence results of positive solutions are obtained by using the monotone iterative method.

In [16], by using Krasnosel'skii’s fixed point theorem, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:

where , , , , and , are the Riemann–Liouville fractional derivatives.

In [17], by using the monotone iterative method, Tian et al. obtained the existence of positive solutions for a boundary value problem of fractional differential equations with p-Laplacian operator as follows:

where , , and , , , and is the Riemann–Liouville fractional derivative.

In [18], by means of the p-Laplacian operator, Han et al. obtained the existence of positive solutions for the boundary value problem of fractional differential equation as follows:

where is continuous, and , are the Riemann–Liouville fractional derivatives.

Based on the above research, this paper analyzed the following fractional three-point boundary value problem with the p-Laplacian operator:

where , and .

The aim is to establish some existence and multiplicity results of positive solutions for BVP (5). This paper is organized as follows. In Section 2, some properties of Green’s function will be given, which are needed later. In Section 3, the existence of multiplicity results of positive solutions of BVP will be discussed (5).

2. Preliminary Knowledge and Lemmas

Lemma 1. (see [14]). Assume that with a fractional derivative of order . Then,

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

Lemma 2. If then the fractional boundary value problem is

The unique solution is where ,

Proof. The general solution to the problem (7) is

From the boundary value condition of (7), ,

Therefore,

Lemma 3. If then the fractional boundary value problem is

The unique solution is where

Proof. Problem (12) is equivalent to

From the boundary value condition of (12), and ; then,

Therefore,

Based on Lemma 2,

Lemma 4. The properties of and are

(i), for (ii), for (iii)(see [13]). , for (iv)(see [13]). , for

Proof.

(i)For , it is easy to show that ; for ,

 So, for ,

 Then, for , .(ii)For , it is easy to show that ; for ,

and for ,

Lemma 5. (see [20]). Let be a Banach space and let be a cone in E. Assume and are open subsets of E with and and let be a continuous and completely continuous. In addition, suppose either

(1), , and , , or(2), , and ,

Lemma 6. (see [16]). Let K be a cone in a real Banach space E, be nonnegative continuous concave functional on K such that , and

Suppose is completely continuous and there exist constants such that

(i) and for (ii) for (iii) for with

Then, has at least three fixed points , and with , , and with .

3. Main Results

When , any , , then E is a real Banach space. is a cone, which can be defined as . Defining the operator , for any ,

and for convenience, the following notation is introduced:

Theorem 1. If there are two positive numbers such that the following conditions hold:

 , for  , for

then the fractional three-point boundary value problem (5) has at least one positive solution and .

Proof. From the continuity of G, H, f, it can be concluded that is continuous. For , , by Lemma 4, we have

It means that . Therefore, the Arzela–Ascoli theorem can prove that the operator is completely continuous.
Let , for . From Lemma 4 and , we can conclude that

Then, when , .
Let , for . Then, we also can conclude from Lemma 4 and that

Therefore, for , . In summary, by Lemma 5, the fractional three-point boundary value problem (5) has at least one positive solution and .

Theorem 2. If there exist positive real numbers such that the following conditions hold:

 , for  , for  , for  then the fractional three-point boundary value problem (5) has at least three positive solutions , , and with

Proof. Firstly, if , then we may assert that is a completely continuous operator. To see this, suppose ; then, . It follows from Lemma 4 and (B₃) that

Therefore, . This together with Lemma 5 implies that is a completely continuous operator. In the same way, if , then assumption (B₅) yields . Hence, condition (ii) of Lemma 6 is satisfied.
To check condition (i) of Lemma 6, we let for . It is easy to verify that and , and so

Thus, for all , we have that for and .
From Lemma 4 and (B₄), one has

This shows that condition (i) of Lemma 6 holds.
Secondly, we verify that (iii) of Lemma 6 is satisfied. By Lemma 6, we have

for with , which shows that condition (iii) of Lemma 6 holds.
To sum up, all the conditions of Lemma 6 are satisfied; from Lemma 6, it follows that there exist three positive solutions , , and with

4. Conclusion

The existence of solutions to three-point boundary value problems of fractional differential equations with the p-Laplacian operators is discussed by using the fixed point exponential theorem and fixed point theorem of cone compression and cone tension. By extending the existence of solutions to boundary value problems, we have obtained the sufficient condition that the boundary value problem has multiple positive solutions or at least one positive solution.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by 2019 Project of Foundational Research Ability Enhancement for Young and Middle-Aged University Faculties of Guangxi (2019KY1046), Nature and Science Foundation of Anhui (2008085QA08), Scientific Research Projects of Institute of Information Technology of GUET (B201911), and Science and Technology Research Project of Heilongjiang Provincial Department of Education (12543079).

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