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

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 getSimilarly, 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 implyThen, 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).