Journal of Function Spaces
Volume 2018 (2018), Article ID 1462825, 6 pages
https://doi.org/10.1155/2018/1462825

Research Article

Existence and Nonexistence of Positive Solutions for Mixed Fractional Boundary Value Problem with Parameter and -Laplacian Operator

Ying Wang

School of Mathematics and Statistics, Linyi University, Linyi 276000, Shandong, China

Correspondence should be addressed to Ying Wang; lywy1981@163.com

Received 23 April 2018; Accepted 10 June 2018; Published 3 September 2018

Academic Editor: Hugo Leiva

Copyright © 2018 Ying 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.

This paper mainly studies a class of mixed fractional boundary value problem with parameter and -Laplacian operator. Based on the Guo-Krasnosel’skii fixed point theorem, results on the existence and nonexistence of positive solutions for the fractional boundary value problem are established. An example is then presented to illustrate the effectiveness of the results.

1. Introduction

In this paper, we consider the existence and nonexistence of positive solution for the following mixed fractional boundary value problem (BVP):

where is a parameter, , , , is the Riemann-Liouville derivative operator, is the Caputo fractional derivative operator, is the -Laplacian operator defined by , , , , , and and satisfies , and is a continuous function.

The theory of fractional differential equation has gained interesting by many researchers due to its deep real world background and, in recent years, more and more papers concern the boundary value problems for fractional-order differential equations; see [1–27]. In [28], Wang, Xiang and Liu use Krasnoselskii’s fixed point theorem and Leggett-Williams theorem to obtain the existence results of BVP, which is given in the following:

where , , , and are Riemann-Liouville derivative operators, is the -Laplacian operator defined by , and is a continuous function.

Lu et al. in [29] investigated the BVP

where , , and are Riemann-Liouville derivative operators, is the -Laplacian operator defined by , and is a continuous function. By the use of Guo-Krasnosel’skii fixed point theorem, the existence and multiplicity results of BVP (3) are obtained.

In this paper, we study the existence and nonexistence of positive solutions for the mixed fractional boundary value problem as BVP (1), which leads to lots of difference and new features. On one hand, compared to the papers mentioned above, which only involve one derivative, our study involves both the Riemann-Liouville fractional derivative and the Caputo fractional derivative, which making the studied problems difficult. On the other hand, under different combinations of superlinearity and sublinearity of the function , results on the existence and nonexistence of positive solutions are received and the impact of the parameter on the existence and nonexistence of positive solutions is also obtained.

2. Preliminaries and Lemmas

Definition 1 (see [30, 31]). The Caputo fractional-order derivative of orders and , , is defined as

where , , denotes the natural number set, , and denotes the integer part of .

Definition 2 (see [30, 31]). Let and let be piecewise continuous on and integrable on any finite subinterval of . Then for , we call

the Riemann-Liouville fractional integral of of order .

Lemma 3 (see [30, 31]). Let , . Then

where and is the smallest integer greater than or equal to .

Let ; then , and we now consider the following BVP:

Lemma 4 (see [32]). If , then BVP (7) has a unique solution

where

For any given , consider the following BVP:

By analysis, we know that (11) can be decomposed into the BVP (7) and the BVP

Lemma 5. If , then BVP

has a unique solution

where

Proof. By Lemma 3, BVP (13) is equivalent to the following integral equation:

Conditions imply that That is,

By , we get

Combining (16) and (18), we can obtain (14). The proof is completed.

Lemma 6. The Green functions and defined separately by (9) and (15) have the following properties:
(1) and are continuous.
(2) .

Proof. Obviously, (1) holds, in the following, and we proof (3). From the definition of , for , we know that (3) holds.
For , we have and then

so we know that . It is also defined by , and we obtain that . Thus, we get that (3) holds. The proof is completed.

Let ; then is a Banach space with the norm For any , , denote

where and then is a positive cone in . Define an integral operator by

We know that is a positive solutions of BVP (1) if and only if is a fixed point of in .

Lemma 7. is a completely continuous operator.

Proof. By the routine discussion, is well defined and we only prove . For any , , by Lemma 6, we have

and then

On the other hand, for , by Lemma 6, we get

By (23) and (24), we can prove that Therefore, we have .
According to the Ascoli-Arzela theorem and the continuity of , we get that is completely continuous. The proof is completed.

Lemma 8 (see [33]). Let be a positive cone in a Banach space; , , and are bounded open sets in , , and , is a completely continuous operator. If the conditions

 , , , and ,

or

 , , , and ,

are satisfied, then has at least one fixed point in .

3. Main Results

Denote

3.1. Existence of BVP (1)

Theorem 9. Assume ; then BVP (1) has at least one positive solution for

where we impose , if , and , if .

Proof. For any satisfying (26), there exists such that

By the definition of , there exists such that

Let . For any , , by the definition of , we know that

Thus, for any , by (28) and (29), we have

Hence, for any , by Lemmas 6 and (30), we conclude that

Then, we have

On the other hand, by the definition of , there exists such that

Choose . Let . For any , by the definition of , we have

For any , by (33) and (34), we have

Then, for any , by Lemma 6 and (35), we have

It follows from the above discussion, (32), (36), and Lemmas 7 and 8, for any , that has a fixed point , so BVP (1) has at least one positive solution ; moreover satisfies . The proof is completed.

By the similar proof as Theorem 9, the following Theorem 10 holds.

Theorem 10. Assume that ; then BVP (1) has at least one positive solution for , where we impose , if , and , if .

3.2. Nonexistence of BVP (1)

Theorem 11. Assume that and ; then there exists , such that for, , BVP (1) has no positive solution.

Proof. By and , there exist positive constants and and and , such that

Setting

we get

Assume that is a positive solution of BVP (1); we will show that it leads to a contradiction. Define , and since , we conclude that

Then, we have , which is a contradiction. Therefore, BVP (1) has no positive solution for . The proof is completed.

By the similar proof as Theorem 11, the following Theorem 12 holds.

Theorem 12. Assume that , , for , and ; then there exists , such that, for , BVP (1) has no positive solution.

4. Examples

Consider the BVP

where is a parameter, , and Let ; choose , andthen , . By Theorem 9, BVP (41) has at least one positive solution for .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The author was supported financially by the National Natural Science Foundation of China (11701252, 11671185), a Project of Shandong Province Higher Educational Science and Technology Program (J16LI03), the Science Research Foundation for Doctoral Authorities of Linyi University (LYDX2016BS080), the Natural Science Foundation of Shandong Province of China (ZR2018MA016), and the Applied Mathematics Enhancement Program of Linyi University.

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