Journal of Function Spaces
Volume 2019 (2019), Article ID 9237856, 10 pages
https://doi.org/10.1155/2019/9237856

Research Article

Existence and Nonexistence of Positive Solutions for Fractional Three-Point Boundary Value Problems with a Parameter

Yunhong Li and Weihua Jiang

School of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018 Hebei, China

Correspondence should be addressed to Yunhong Li; mathhong@126.com

Received 13 October 2018; Accepted 19 December 2018; Published 3 January 2019

Academic Editor: Maria Alessandra Ragusa

Copyright © 2019 Yunhong Li and Weihua Jiang. 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 work, we investigate the existence and nonexistence of positive solutions for p-Laplacian fractional differential equation with a parameter. On the basis of the properties of Green’s function and Guo-Krasnosel’skii fixed point theorem on cones, the existence and nonexistence of positive solutions are obtained for the boundary value problems. We also give some examples to illustrate the effectiveness of our main results.

1. Introduction

Fractional calculus has played an important role in the modeling of different physical and natural phenomena, such as fluid mechanics, control system, and many other branches of engineering. In recent years, there are many papers concerning the existence of positive solutions for nonlinear fractional differential equations; see [1–18] and the references cited therein.

In [1], Han et al. studied the existence of positive solutions for the following problems with the generalized p-Laplacian operator:

On the basis of the properties of Green’s function and Guo-Krasnosel’skii fixed point theorem on cones, some new existence results of at least one or two positive solutions are obtained by different eigenvalue interval for the aforementioned boundary value problems.

Xu et al. [2] showed the existence of multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

where and is the standard Riemann-Liouville fractional derivative. By means of the Leray-Schauder nonlinear alternative and a fixed point theorem on cones, they deduce multiple positive solutions to singular positive and semipositive m-point boundary value problems.

Shen et al. [3] investigate the following fractional thermostat model with a parameter:

where , and is the Caputo’s fractional derivatives. Using the fixed point theorem on cones, the existence and nonexistence results for positive solutions are discussed for the boundary value problems.

Motivated by the aforementioned works, this paper is concerned with the positive solutions for p-Laplacian fractional differential equation with a parameter

where . and are the Riemann-Liouville fractional derivatives, , and . Based on the properties of Green’s function and Guo-Krasnosel’skii fixed point theorem on cones, the existence of positive solutions are obtained for problems (4)-(8).

We will always suppose the following conditions are satisfied:

and ;

is continuous.

2. Background and Definitions

For the convenience of the reader, we state some basic definitions and lemmas about fractional calculus theory, which can be found in [19, 20].

Definition 1. The fractional integral of order of a function is given by

provided that the right side is pointwise defined on , where

Definition 2. For a continuous function , the Riemann-Liouville derivative of fractional order is defined as

where , provided that the right side is pointwise defined on .

Lemma 3. Let . Assume that . Then

as the unique solution, where is the smallest integer greater than or equal to .

In order to obtain our main results, we need the following Guo-Krasnoselskii fixed point theorem in [21].

Theorem 4. Let be a Banach space, and be a cone in . Assume that and are open subsets of such that . If

is a completely continuous operator such that either
(i) if , and , or
(ii) if , and ,
then has a fixed point in .

3. Preliminary Lemmas

Lemma 5. The boundary value problems (4)-(8) are equivalent to the following equation:

where

is the inverse function of , a.e., and .

Proof. According to Lemma 3, (4) is equivalent to the following integral equation:

Conditions (5) imply that i.e.,

So,

From (6), we have

By use of (18) and (20), we get

Therefore,

In view of Lemma 3, we have

Conditions (7) imply that i.e.,

So,

From (8), we get

By use of (24) and (26), we can obtain

The proof is complete.

Lemma 6. Functions and defined by (15) and (16), respectively, are continuous on and satisfy
(i) , for ;
(ii) , for ;
(iii) , for , where

Proof. If , then

If , then
If , then

So we get if , then

where Obviously,
If , then

If , then

If , we get

Consequently, we get

Similarly, we can get

where
The proof is complete.

Let be the real Banach space with the maximum norm and define the cone by

where

Define the operator on by

Lemma 7. is completely continuous.

Proof. For any , in view of (26) and Lemma 6, we get

So we obtain

Therefore, . In view of continuity of and , we have is continuous.
Let be bounded, i.e., there exists a positive constant such that , for all , let , then, for and , we get

Hence, is uniformly bounded.
On the other hand, since is continuous on , it is uniformly continuous on . Thus, for any , there exists a constant , such that with imply

Then, for all ,

Hence, is equicontinuous. By Arzela-Ascoli theorem, we have is completely continuous.

4. Main Results

For convenience we introduce the following notations. Let

The following theorems are the main results in this paper.

Theorem 8. If and hold, and

then for each

the boundary value problems (4)-(8) have at least one positive solution. Here we impose if and if

Proof. Suppose holds, we may choose , so that, for each , there are . Thus, if and , then by (40) and Lemma 6, we have

Let , then the previous inequality show that for .
On the other hand, suppose (A2) holds, there is such that , for . Thus, if and , by (40) and Lemma 6, we have

Let , then the previous inequality show that for .
Thus, from Theorem 4, we know that the operator has a fixed point in .

Theorem 9. If and hold, and

then for each

the boundary value problems (4)-(8) have at least one positive solution. Here we impose if and if

Proof. Suppose holds, we may choose so that, for each there are . Thus, if and , then by (40) and Lemma 6, we have

Let , then the previous inequality shows that for .
Suppose (A4) holds, we consider two cases.
Case 1. Suppose is bounded, then there exists some , such that , for . Thus, if and if

by (40) and Lemma 6, we have

Case 2. Suppose is unbounded; there is such that , for . Then there exists some , such that , for .
Thus, if and , by (40) and Lemma 6, we have

Let , then the previous inequality shows that for . Thus, from Theorem 4, we know that the operator has a fixed point in .

Theorem 10. If and , then there exists such that for all , the boundary value problems (4)-(8) have no positive solution.

Proof. Since and , there exist positive constants , such that and
for ;
for .
Let

then we have

Assume is a positive solution of the boundary value problems (4)-(8); let

then for all , we get

which is a contraction. Therefore, the boundary value problems (4)-(8) have no positive solution.

Theorem 11. If and , then there exists such that, for all , the boundary value problems (4)-(8) have no positive solution.

Proof. Since and , there exist positive constants , such that and
for ,
for .
Let

then we have

Suppose is a positive solution of the problems (4)-(8); let

then for all , we get

which is a contraction. Therefore, the boundary value problems (4)-(8) have no positive solution.

5. Examples

In this section, we give some simple examples to explain our results.

Example 1. For the problems (4)-(8), let , then we get Let

then From a direct calculation, we get

In view of Theorem 8, we get that problems (4)-(8) have at least one positive solution when .

Example 2. For the problems (4)-(8), let , then we get Let

then From a direct calculation, we get

In view of Theorem 8, we get that problems (4)-(8) have at least one positive solution when .

6. Conclusions

The fixed point theorem on cones is used to solve the three-point boundary value problems of a kind of nonlinear fractional differential equation with a parameter. Under certain conditions of the nonlinearity, the existence and nonexistence of positive solutions are obtained for the boundary value problems when the parameter belongs to different intervals.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Foundation of Hebei Education Department (QN2018104) and Hebei Province Natural Science Fund (A2018208171).

References

Z. Han, H. Lu, and C. Zhang, “Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian,” Applied Mathematics and Computation, vol. 257, pp. 526–536, 2015.

X. Xu and H. Zhang, “Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations,” Boundary Value Problems, vol. 34, pp. 1–18, 2018.

C. Shen, H. Zhou, and L. Yang, “Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter,” Mathematical Methods in the Applied Sciences, vol. 39, no. 15, pp. 4504–4511, 2016.

W. Jiang and N. Kosmatov, “Solvability of a third-order differential equation with functional boundary conditions at resonance,” Boundary Value Problems, vol. 81, pp. 1–20, 2017.

J. R. Webb, “Existence of positive solutions for a thermostat model,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 923–938, 2012.

T. Chen and W. Liu, “An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1671–1675, 2012.

S. K. Ntouyas and S. Etemad, “On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions,” Applied Mathematics and Computation, vol. 266, pp. 235–243, 2015.

Y. Wang, L. Liu, and Y. Wu, “Extremal solutions for p-Laplacian fractional integro-differential equation with integral conditions on infinite intervals via iterative computation,” Advances in Difference Equations, vol. 24, pp. 1–14, 2015.

G. Chai, “Positive solutions for boundary value problem of fractional differential equation with p-Laplacian Operator,” Boundary Value Problems, vol. 2012, pp. 1–20, 2012.

X. Zhang and M. Feng, “Existence of a positive solution for one-dimensional singular p-Laplacian problems and its parameter dependence,” Journal of Mathematical Analysis and Applications, vol. 413, pp. 566–582, 2014.

X. Feng, H. Feng, and H. Tan, “Existence and iteration of positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian,” Applied Mathematics and Computation, vol. 266, pp. 634–641, 2015.

A. Cabada and Z. Hamdi, “Nonlinear fractional differential equations with integral boundary value conditions,” Applied Mathematics and Computation, vol. 228, pp. 251–257, 2014.

Y. Li and A. Qi, “Positive solutions for multi-point boundary value problems of fractional differential equations with p-Laplacian,” Mathematical Methods in the Applied Sciences, vol. 39, no. 6, pp. 1425–1434, 2016.

Y.-Y. Yang and Q.-R. Wang, “Multiple positive solutions for p-Laplacian equations with integral boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 453, no. 1, pp. 558–571, 2017.

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.

H. Qu and X. Liu, “Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance,” Boundary Value Problems, vol. 127, pp. 1–10, 2013.

A. Jannelli, M. Ruggieri, and M. P. Speciale, “Analytical and numerical solutions of time and space fractional advection–diffusion–reaction equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 70, pp. 89–101, 2019.

A. Jannelli, M. Ruggieri, and M. P. Speciale, “Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries,” Nonlinear Dynamics, vol. 92, no. 2, pp. 543–555, 2018.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, London, UK, 1964.