Journal of Function Spaces
Volume 2018 (2018), Article ID 7351653, 10 pages
https://doi.org/10.1155/2018/7351653

Research Article

Positive Solutions for a System of Nonlinear Semipositone Boundary Value Problems with Riemann-Liouville Fractional Derivatives

Xiaowei Qiu,¹ Jiafa Xu,¹ Donal O’Regan,² and Yujun Cui³

¹School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
²School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
³State 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 Jiafa Xu; xujiafa292@sina.com

Received 18 January 2018; Accepted 28 March 2018; Published 6 May 2018

Academic Editor: Gennaro Infante

Copyright © 2018 Xiaowei Qiu et al. 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.

We study the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives , , , and where is a real number and is the standard Riemann-Liouville fractional derivative of order . Under some appropriate conditions for semipositone nonlinearities, we use the fixed point index to establish two existence theorems. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities.

1. Introduction

In this paper, we investigate the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives

where is a real number and is the standard Riemann-Liouville fractional derivative of order . The nonlinear terms , are bounded below; that is, satisfy the following.

(H1) there exists a real number , such that ,

Existence and multiplicity of solutions for fractional differential equations are widely studied in the literature; see [1–14] and the references therein. For example, in [1], the authors used the Guo-Krasnosel’skii fixed point theorem to investigate the existence of positive solutions for the singular fractional differential system

where satisfy

Condition (3) is used to study various types of fractional systems (see [1–12] and the references therein).

In this paper we use the fixed point index to study the existence of positive solutions for the system of nonlinear semipositone fractional boundary value problem (1). Under some appropriate conditions for , we use the fixed point index to obtain our results. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities (see [13–15]), which depend on the unknown functions and their derivatives , .

2. Preliminary

Definition 1 (see [16, 17]). The Riemann-Liouville fractional derivative of order of a continuous function is given by

where with denoting the integer part of a number , provided that the right hand side is pointwise defined on .
We first study the Green functions of problem (1). Let

Then we have

Lemma 2. Let be as in (1). Then (1) is equivalent to

which takes the form

Let , . Then an argument similar to that in [18, Lemma 2.7] and [19, Lemma 3] establishes the result (we omit the standard details).

Lemma 3 ([19, Lemma 4]). The functions Moreover, the following inequalities are satisfied:

Lemma 4 ([19, Lemma 5]). Let for all Let

Then

Lemma 5. (i) If is a positive solution of (7), then is a positive solution of the following differential equation:

where

and are continuous, and

(ii) If is a solution of (13) and , , then is a positive solution of (7).

Proof. If is a positive solution of (7) then (note ) we obtain and

Similarly, we have

that is, satisfies (13). Therefore, (i) holds. Similarly, it is easy to prove (ii). This completes the proof.

From Lemma 5, to obtain a positive solution of (7), we only need to find solutions , of (13) satisfying , , If , are solutions of (13), then , satisfy

Let , , Then is a real Banach space, and is a cone on . We denote for Now, note that , solve (1) if and only if , are fixed points of operator

Therefore, if is a positive fixed for with , for , then is a positive solution for (1). Moreover, from the continuity of and , we know that , are continuous and completely continuous operators.

Lemma 6. Let , Then is a cone in and

Proof. From (9) for we have

Also from (9) and the above inequality, for every , we obtain

for all . Similarly, . Therefore This completes the proof.

To obtain a positive solution of (1), we seek a positive fixed point of with , (note mean that , for ). From Lemma 6, we have . For we have

As a result, for if Similarly, if , we have , for .

Lemma 7 (see [20]). Let be a bounded open set and a continuous or completely continuous operator. If there exists such that for all and , then , where denotes the fixed point index on .

Lemma 8 (see [20]). Let be a bounded open set with . Suppose is a continuous or completely continuous operator. If for all and , then

3. Main Results

Let , In the sequel, we use and to stand for different positive constants. Now, we list our assumptions on .

(H2) There exist such that

(i) is concave and strictly increasing on (and );(ii)there exist , , for all such that

(iii) for

(H3) For all , there is a constant such that

(H4) There exist , such that

(i) is convex and strictly increasing on (and );(ii)for all ,

(iii)there exist such that , for .

(H5) There exist , , , for all , such that

where

Theorem 9. Suppose that (H1)–(H3) hold. Then (1) has at least one positive solution.

Proof. We first prove that there exists such that

where is a given function. Suppose there exist , with , then , for . From (i), (ii) of (H2) we have

From (ii) of (H2) we have

From (30) and (i) of (H2) we obtain

This together with (iii) of (H2) yields

Then (32) is substituted into (29) and we obtain

Multiplying by for (33) and integrating over , we have

using the fact that

which can be derived from (12) in Lemma 4. From (34) we obtain

Note that (note that and from Lemma 6 and ) and we have

From (29) we have

Multiplying by and integrating over we obtain

Consequently, we have

Note that we may assume for Then and . For , we have

Hence, . Note , and thus there exists such that . Therefore if , with then and . Thus if we take then (28) is true. Lemma 7 implies

Let . From (H3) we have

so . Similarly . Hence for Thus

It follows from Lemma 8 that

From (42) and (45) we have

Therefore the operator has at least one fixed point in and so (1) has at least a positive solution. This completes the proof.

Theorem 10. Suppose that (H1), (H4), and (H5) hold. Then (1) has at least one positive solution.

Proof. We first show that there exists such that

Suppose there exist , with , then , for From (i), (ii) of (H4), we have

From (ii) of (H4), we get

From (49) and (i), (iii) of (H4) we have

Then substitute (50) into (48) and we obtain

Multiplying by for (51) and integrating over , from (35), we obtain

Consequently, we have

Note that (note and ) and we have

From (50) and Lemma 3 we have

so is bounded. Note , and thus there exists such that Therefore if with then and . Thus if we take then (47) is true. Lemma 8 implies

Let and consider It follows from (H5) that

and, hence, . Similarly for . Therefore for all Thus

It follows from Lemma 7 that

From (56) and (59), we have

Therefore the operator has at least one fixed point on and so (1) has at least one positive solution, which completes the proof.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant no. 11601048), Natural Science Foundation of Chongqing (Grant no. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJ1703050), and Natural Science Foundation of Chongqing Normal University (Grant nos. 16XYY24, 15XLB011).

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