Journal of Function Spaces
Volume 2018 (2018), Article ID 1547293, 7 pages
https://doi.org/10.1155/2018/1547293

Research Article

Existence of Uniqueness and Nonexistence Results of Positive Solution for Fractional Differential Equations Integral Boundary Value Problems

Yongqing Wang

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

Correspondence should be addressed to Yongqing Wang; wyqing9801@163.com

Received 14 April 2018; Revised 20 July 2018; Accepted 17 October 2018; Published 4 December 2018

Academic Editor: Yong H. Wu

Copyright © 2018 Yongqing Wang. 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 class of fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are obtained by means of the iterative technique. The interesting point lies in that the assumption on nonlinearity is closely associated with the spectral radius corresponding to the relevant linear operator.

1. Introduction

In this paper, we consider the existence of uniqueness and nonexistence of positive solution for the following fractional differential equations:

where , , , , , is the standard Riemann-Liouville derivative, is continuous, and is nonnegative.

In the recent years, many results were obtained to deal with the existence of solutions for nonlinear differential equations by using nonlinear analysis methods; see [1–16] and references therein. The fractional nonlocal boundary value problems have particularly attracted a great deal of attention (see [17–27]). While there are a lot of works dealing with the existence and multiplicity of solutions for nonlinear fractional differential equations, the results dealing with the uniqueness of solution are relatively scarce (see [28–35]). The main tool used in most of the papers dealing with the uniqueness of solution is the Banach contraction map principle provided that the nonlinearity is a Lipschitz continuous function. When , and is continuous on , Zhang and Zhong [34] established the uniqueness results of solution to problem (1) by using the Banach contraction map principle. It is worth mentioning that only positive solutions are meaningful in most practical problems. As far as we know, the nonexistence of positive solution has seldom been considered up to now.

Motivated by the above work, the aim of this paper is to establish the existence of uniqueness and nonexistence of positive solution to problem (1). Our analysis relies on the iterative technique on the cone derived from the properties of the Green function. This article provides some new insights. Firstly, the uniqueness results are obtained under some conditions concerning the spectral radius with respect to the relevant linear operator. In addition, the error estimation of the iterative sequences is given. Secondly, we impose weaker positivity conditions on ; that is, the Lipschitz constant is generalized to a function and may be singular at . Finally, the nonexistence results of positive solution are obtained under conditions concerning the spectral radius of the relevant linear operator.

2. Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory and lemmas.

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

provided that the right-hand side is point-wise defined on .

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

where ; denotes the integer part of number , provided that the right-hand side is point-wise defined on .

For convenience, we here list the assumptions to be used throughout the paper.

   is nonnegative, and

   is continuous.

Lemma 3 ([34]). For any , the unique solution of the boundary value problem

is

where

Lemma 4. The function has the following properties:

(1);(2);(3).

Proof. It is obvious that (1) and (2) hold. In the following, we will prove (3).
Case (i) (). Noticing , we have

which implies

Noticing , we have

On the other hand, it follows from

that

Therefore,

Case (ii) (). It is easy to see that

On the other hand, we have

It follows from (10), (13), (14), and (15) that (3) holds.

Lemma 5. The function has the following properties:

(1);(2);(3),

where

Proof. It can be directly deduced from Lemma 4 and the definition of , so we omit the proof.

Let be endowed with the maximum norm , . Define cones , by

It is clear that is nonempty set since .

For convenience, we list here one more assumption to be used later:

There exists such that

Moreover,

Define operators and as follows:

Lemma 6. Assume that holds; then .

Proof. It is clear that ,  . For any , we have

Then is well-defined on . It follows from Lemma 5 that .

By virtue of the Krein-Rutmann theorem and Lemma 5, we have the following lemma.

Lemma 7. Assume that holds. Then is a completely continuous linear operator. Moreover, the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue ; that is, .

3. Main Results

Theorem 8. Assume that holds. Then (1) has a unique positive solution if the spectral radius .

Proof. It follows from that is not a fixed point of . Then we only need to prove that has a unique fixed point in .
Firstly, we will prove has a fixed point in .
For any , let

Then

By Lemma 7, we have that . It is easy to see that

For any , set

We may suppose that (otherwise, the proof is finished). It follows from (23) and (24) that

Then, we have

By induction, we can get

Then, for any , one has

It follows from that

which implies is a Cauchy sequence. Therefore, there exists such that converges to . Clearly is a fixed point of .
In the following, we will prove the fixed point of is unique.
Suppose is a positive fixed point of . Then there exists such that

Therefore,

By induction, we can get

It follows from that

which implies the positive fixed point of is unique.

Remark 9. The unique positive solution of (1) can be approximated by the iterative schemes: for any , let

and then . Furthermore, we have error estimation

and with the rate of convergence

Remark 10. The spectral radius satisfies and . Particularly,

Theorem 11. Assume that the following condition holds:
There exists satisfying

such that

Then (1) has no positive solution if the spectral radius , where

Proof. We only need to prove that has no fixed point in . Otherwise, there exists , such that .
By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying

It is clear that . Therefore, there exists such that

It follows from that . It is obvious that is increasing on . By induction, we can get , . Thus,

Noticing , we have , which contradicts with .

Theorem 12. Assume that the following condition holds:
There exists satisfying

such that

Then (1) has no positive solution if the spectral radius , where

Proof. Suppose that there exists , such that .
By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying

It is clear that . Therefore, there exists such that

It follows from that . Noticing that is increasing on , by induction, we have , . Thus,

It follows from that , which contradicts with .

4. Example

Example 1. Consider the following integral boundary value problem:

with

where

By direct calculations, we have

It is clear that and hold. Clearly, we have

Let ; then we have

It is easy to get that holds.

Denote

By direct calculations, we have

, we have

Therefore,

It is obvious that

and

which implies that

Lemma 7 and Remark 10 can guarantee that

So all of the assumptions of Theorem 8 are satisfied. As a result, BVP (51) has a unique positive solution.

Example 2. Consider BVP (51) with

Clearly, we have

It is not difficult to check that holds.

It follows from Example 1 that

So all of the assumptions of Theorem 11 are satisfied. As a result, BVP (51) has no positive solution.

5. Conclusions

In this paper, we consider the existence of positive solution for fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are established under conditions closely associated with the spectral radius with respect to the relevant linear operator. In addition, the unique positive solution can be approximated by an iterative scheme, and the error estimation of the iterative sequences is also given.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA036) and the National Natural Science Foundation of China (11571296).

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