## Graph-Theoretic Techniques for the Study of Structures or Networks in Engineering

View this Special IssueResearch Article | Open Access

Dong Li, Yang Liu, Chunli Wang, "Multiple Positive Solutions for Fractional Three-Point Boundary Value Problem with *p*-Laplacian Operator", *Mathematical Problems in Engineering*, vol. 2020, Article ID 2327580, 6 pages, 2020. https://doi.org/10.1155/2020/2327580

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

**Academic Editor:**Jia-Bao Liu

#### 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 [1â€“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 [7â€“10].

Recently, fractional differential equations have gained considerable attention (see [11â€“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 [16â€“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 [21â€“25].

In [15], Chai studied the boundary value problems of fractional differential equations with *p*-Laplacian operator as follows:where , , , , and , , are the standard Riemannâ€“Liouville derivatives. Some existence results of positive solutions are obtained by using the monotone iterative method.

In [16], by using Krasnosel'skiiâ€™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 Riemannâ€“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 Riemannâ€“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 Riemannâ€“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 Greenâ€™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) isFrom 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 toFrom 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 ,â€‰So, for ,â€‰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:*â€‰* , for *â€‰* , 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 haveIt means that . Therefore, the Arzelaâ€“Ascoli theorem can prove that the operator is completely continuous.

Let , for . From Lemma 4 and , we can conclude thatThen, when , .

Let , for . Then, we also can conclude from Lemma 4 and thatTherefore, 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:*â€‰* , for *â€‰* , for *â€‰* , for *â€‰*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 (*B*_{3}) thatTherefore, . This together with Lemma 5 implies that is a completely continuous operator. In the same way, if , then assumption (*B*_{5}) 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 soThus, for all , we have that for and .

From Lemma 4 and (*B*_{4}), one hasThis shows that condition (i) of Lemma 6 holds.

Secondly, we verify that (iii) of Lemma 6 is satisfied. By Lemma 6, we havefor 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).

#### References

- K. B. Oldham and J. Spanier,
*The Fractional Calculus (Theory and Applications of Differentiation and Integration to Arbitrary Order)*, Academic Press, San Diego, CA, USA, 1974. - S. G. Kilbas and A. A. Marichev,
*Fractional Integrals and Derivatives (Theory and Applications)*, Gordon & Breach, New York, NY, USA, 1993. - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, Wiley, New York, NY, USA, 1993. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, CA, USA, 1999. - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, New Jersey, NJ, USA, 2000. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, Netherlands, 2006. - V. Lakshmikantham, S. Leela, and J. Vasundhara,
*Theory of Fractional Dynamic Systems*, Cambridge Academic Publishers, Cambridge, UK, 2009. - R. Herrmann,
*Fractional Calculus (An Introduction for Physicists)*, World Scientific, Singapore, 2014. - T. M. Atanackoviâ€™c, S. Pilipoviâ€™c, B. Stankoviâ€™c, and D. Zorica,
*Fractional Calculus with Applications in Mechanics*, Wiley, New York, NY, USA, 2014. - Y. Liu, D. P. Xie, D. D. Yang, and C. Z. Bai, â€śTwo generalized Lyapunov-type inequalities for a fractional
*p*-Laplacian equation with fractional boundary conditions,â€ť*Journal of Inequalities and Applications*, vol. 98, no. 1, pp. 1â€“16, 2017. View at: Publisher Site | Google Scholar - L. S. Zhang, X. Wu, and Y. Caccetta, â€śExtremal solutions for
*p*-Laplacian differential systems via iterative computation,â€ť*Applied Mathematics Letters*, vol. 26, no. 12, pp. 1151â€“1158, 2013. View at: Publisher Site | Google Scholar - S. N. Rao, M. Singh, and M. Z. Meetei, â€śMultiplicity of positive solutions for Hadamard fractional differential equations with
*p*-Laplacian operator,â€ť*Boundary Value Problems*, vol. 43, no. 1, pp. 1â€“25, 2020. View at: Publisher Site | Google Scholar - Y. Wang, â€śMultiple positive solutions for mixed fractional differential system with
*p*-Laplacian operators,â€ť*Boundary Value Problems*, vol. 144, no. 1, pp. 1â€“17, 2019. View at: Publisher Site | Google Scholar - Y. He and B. Bi, â€śExistence and iteration of positive solution for fractional integral boundary value problems with
*p*-Laplacian operator,â€ť*Advances in Difference Equations*, vol. 415, pp. 1â€“15, 2019. View at: Publisher Site | Google Scholar - G. Chai, â€śPositive solutions for boundary value problem of fractional differential equation with
*p*-Laplacian operator,â€ť*Boundary Value Problems*, vol. 18, no. 1, pp. 1â€“15, 2012. View at: Publisher Site | Google Scholar - Y. Tian, Y. Wei, and S. Sun, â€śMultiplicity for fractional differential equations with
*p*-Laplacian,â€ť*Boundary Value Problems*, vol. 127, no. 1, pp. 1â€“18, 2018. View at: Publisher Site | Google Scholar - Y. Tian, Z. Bai, and S. Sun, â€śPositive solutions for a boundary value problem of fractional differential equation with
*p*-Laplacian operator,â€ť*Advances in Difference Equations*, vol. 349, pp. 1â€“19, 2019. View at: Google Scholar - Z. Han, H. Lu, and C. Zhang, â€śPositive solutions for eigenvalue problems of fractional differential equation with generalized
*p*-Laplacian,â€ť*Applied Mathematics and Computation*, vol. 257, no. 1, pp. 526â€“536, 2015. View at: Publisher Site | Google Scholar - C. Z. Bai, â€śExistence and uniqueness of solutions for fractional boundary value problems with
*p*-Laplacian operator,â€ť*Advances in Difference Equations*, vol. 4, pp. 1â€“12, 2018. View at: Google Scholar - Y. H Li, â€śMultiple positive solutions for nonlinear mixed fractional differential equation with
*p*-Laplacian operator,â€ť*Advances in Difference Equations*, vol. 112, no. 1, pp. 1â€“12, 2019. View at: Publisher Site | Google Scholar - J. B. Liu, J. Zhao, and Z. Cai, â€śOn the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks,â€ť
*Physica A*, vol. 540, Article ID 123073, 2020. View at: Publisher Site | Google Scholar - J. B. Liu, J. Zhao, and Z. X. Zhu, â€śOn the number of spanning trees and normalized Laplacian of linear octagonal-quadrilateral networks,â€ť
*International Journal of Quantum Chemistry*, vol. 119, Article ID 25971, 2019. View at: Publisher Site | Google Scholar - B. B. Zhou and L. L. Zhang, â€śExistence of positive solutions of boundary value problems for high-order nonlinear conformable differential equations with
*p*- Laplacian operator,â€ť*Advances in Difference Equations*, vol. 351, no. 1, pp. 1â€“14, 2019. View at: Publisher Site | Google Scholar - K. S. Jong, â€śExistence and uniqueness of positive solutions of a kind of multi-point boundary value problems for nonlinear fractional differential equations with
*p*-Laplacian operator,â€ť*Mediterranean Journal of Mathematics*, vol. 15, no. 3, 2018. View at: Publisher Site | Google Scholar - H. X. Wang and W. M. Hu, â€śExistence of solutions for a class of fractional differential equations with
*p*-Laplacian operators and integral boundary conditions,â€ť*Mathematics in Practice and Theory*, vol. 46, no. 16, pp. 228â€“236, 2016. View at: Google Scholar

#### Copyright

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.