Journal of Function Spaces
Volume 2018 (2018), Article ID 2176809, 9 pages
https://doi.org/10.1155/2018/2176809

Research Article

The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions

Wenjie Ma¹ and Yujun Cui²

¹Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
²State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Yujun Cui; cyj720201@163.com

Received 13 June 2018; Accepted 30 July 2018; Published 12 August 2018

Academic Editor: Liguang Wang

Copyright © 2018 Wenjie Ma and Yujun Cui. 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 eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions , , , , where is Caputo fractional derivative, , and is continuous. By using the Guo-Krasnoselskii’s fixed point theorem on cone and the properties of the Green’s function, some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained.

1. Introduction

The experience of the last few years has fully borne out the fact that the integer order calculus is not as widely used as fractional order calculus in some fields such as chemistry, control theory, and signal processing. On the remarkable survey of Agarwal, Benchohra, and Hamani [1] it is pointed out that fractional differential equations constitute a fundamental tool in the modeling of some phenomena (see also [2–4]). The use of fractional order is more accurate for the description of phenomena, so the study of fractional differential equations becomes the mainstream with the help of techniques of nonlinear analysis. We refer the reader to [5–36] for recent results. For example, in [9], the author studied the following fractional differential equation:

with boundary conditions

where , , and is the Caputo derivative. They solved the above problem by means of classical fixed point theorems.

In [5], the boundary value problem for the following nonlinear fractional differential equation was discussed:

with boundary conditions

where is the Riemann-Liouville differentiation, . By using a fixed point theorem, a new result of the existence of three positive solutions is obtained.

In [15], the authors investigated the following class of BVP:

with boundary conditions

where , , is a given function, and denotes the Caputo differentiation. The author investigated this problem by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green’s function, and Guo-Krasnoselskii fixed point theorem on cone. Similar problems can be referred to in [25].

In this paper, we investigate the eigenvalue problem for Caputo fractional boundary value problem with Riemann-Stieltjes integral boundary conditions

where , is continuous, is a parameter, is the Caputo fractional derivative, and is a bounded variation function with positive measures with

Our proof is based on the properties of the Green’s function and the Guo-Krasnosel’skii fixed point theorem on cone.

2. Preliminaries

In order to solve problem (7), we provide the properties related to problem (7).

Definition 1 (see [3]). The Caputo’s fractional derivative of order for a function is defined as

where is the smallest integer greater than or equal to .

Lemma 2 (see [3]). Let . If we assume , then the fractional differential equation

has the general solution , , , where is the smallest integer greater than or equal to .

Lemma 3 (see [3]). Given that with a fractional derivative of order that belongs to . Then

where is the smallest integer greater than or equal to .

Firstly, we consider the following linear Caputo fractional differential equation:

Lemma 4. Let and assume that . Then is the solution of the above boundary value problem (12), if and only if satisfies the following integral equation:

where

and

Proof. Applying the fractional integral of order to both sides of (12) for , we get the following formula:

According to and Lemma 3, we obtain

where . Since , we deduce that

Therefore,

Substituting the above equality into (17), one has

and

The proof is completed.

Lemma 5. The Green’s function has the following properties:
(i) , for ;
(ii) , for and , where

Proof. (i) Obviously, the inequality holds from the representation of .
(ii) In view of and , we have .
For , we have

For , we have

Thus, the above two inequalities yield the inequality in (ii). The proof is completed.

Let , ; then is a Banach space. We define the cone by

Let be the operator defined as

Thus, the fixed point of the above integral equation is equivalent to the solution of the BVP (7).

Lemma 6. and is a completely continuous operator, where is defined in (26).

Proof. By Lemma 5, for , we have

Hence we have . Let be bounded. Then there exists a constant such that for . Let . Then

Thus, is bounded. Put and . We deduce that

Since are uniformly continuous on , is equicontinuous, by using Arzela-Ascoli’s theorem, we can prove that is completely continuous. The proof is completed.

The following Guo-Krasnoselskii’s fixed point theorem is used to prove the existence of positive solution of (7).

Theorem 7 (see [37]). Let be a cone of real Banach space and let and be two bounded open sets in such that . Let operator be completely continuous operator. If one of the following two conditions holds:
(1) for all , for all ,
(2) for all , for all ,
then has at least one fixed point in .

3. Existence of Positive Solutions

In this section, we investigate the existence of positive solutions for integral boundary value problems of fractional differential equation (7).

For convenience, we denote them by

Theorem 8. Suppose that holds; then for , problem (7) has a positive solution. Here we impose if and if .

Proof. Let and satisfy

According to the definition of , we know that there exists a constant such that

Put . Let . We have and

Therefore, for .
By the definition of , we know that there exists such that

Let , . Then for , by (25) we have

and thus

Therefore, for .
By Theorem 7, if , we assert that has a fixed point in and therefore problem (7) has at least one positive solution. The proof is completed.

Theorem 9. Assume that holds. Then for , the problem (7) has at least a positive solution. Here we impose if and if .

Proof. Let and such that

According to the definition of , there exists a constant such that

Put . Let ; we have

Therefore, for .
It follows from the definition of that there exists a constant such that

This together with the continuity of implies that

holds for some .
Let , . For , we conclude that

Therefore, for .
By Theorem 7, if , we conclude that has a fixed point in , and so problem (7) has one positive solution. The proof is completed.

4. Nonexistence of Positive Solutions

In this section, we present some sufficient conditions for nonexistence of positive solution to integral boundary value problems of fractional differential equation (7).

Theorem 10. If and , then there exists a such that problem (7) has no positive solution for .

Proof. Since , , we have for , and for , where are positive numbers with . Let ; then we have for . Suppose is a positive solution of problem (7); then we are going to prove that this leads to a contradiction for . Since , for , then

which is a contradiction. Therefore this completes the proof.

Theorem 11. If , , for and , then there exists a such that problem (7) has no positive solution for all .

Proof. Since , , we have for , and for , where are positive numbers and . Let ; then we have for . Suppose is a positive solution of problem (7); then we are going to prove that this leads to a contradiction for . Since , then

which is a contradiction. Therefore this completes the proof.

5. Example

Example 1. We consider the following fractional equation:

where , , , . We obtain , , .
It is easy to see that, for all ,

and

Then , ; from Theorem 8, problem (45) has a positive solution.

Example 2. We consider the following fractional equation:

where , , , and . We obtain , , and .
One can easily see that, for all ,

and

Then , ; from Theorem 9, problem (48) has a positive solution.

Example 3. We consider the following fractional equation:

where , , , and . We obtain , , and .
By direct calculation, we obtain that

and

Take and . By Theorem 10, problem (51) has no positive solution for .

Example 4. We consider the following fractional equation:

where , , , and . We obtain , , and .
Obviously, we can infer that

and

Take and . By Theorem 11, problem (54) has no positive solution for .

Data Availability

The dataset supporting the conclusions of this article is included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation (NNSF) of China (11371221, 11571207), Shandong Natural Science Foundation (ZR2018MA011), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.

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