Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 2823971, 12 pages
http://dx.doi.org/10.1155/2016/2823971

Research Article

Existence and Nonexistence of Positive Solutions for Coupled Riemann-Liouville Fractional Boundary Value Problems

Johnny Henderson,¹ Rodica Luca,² and Alexandru Tudorache³

¹Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA
²Department of Mathematics, Gh. Asachi Technical University, 700506 Iasi, Romania
³Faculty of Computer Engineering and Automatic Control, Gh. Asachi Technical University, 700050 Iasi, Romania

Received 12 January 2016; Revised 13 April 2016; Accepted 14 June 2016

Academic Editor: Josef Diblík

Copyright © 2016 Johnny Henderson 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 investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with two parameters, subject to coupled integral boundary conditions.

1. Introduction

Fractional differential equations describe many phenomena in various fields of engineering and scientific disciplines such as physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, and electromagnetics (see [1–5]). Integral boundary conditions arise in thermal conduction problems, semiconductor problems, and hydrodynamic problems (see, e.g., [6–9]).

We consider the system of nonlinear fractional differential equations with parameters

with the coupled integral boundary conditions

where , , , , and denote the Riemann-Liouville derivatives of orders and , respectively, and the integrals from are Riemann-Stieltjes integrals. The boundary conditions include multipoint and integral boundary conditions, as well as the sum of these in a single framework.

Under some assumptions on the nonnegative functions and , we present intervals for the parameters and such that positive solutions of - exist. By a positive solution of problem - we mean a pair of functions satisfying and with for all or for all . The nonexistence of positive solutions for - is also investigated. The existence, multiplicity, and nonexistence of positive solutions ( for all and ) for system with different coupled boundary conditions, namely,

were investigated in [10, 11] (where and are nonnegative and nonsingular functions) and in [12] (where and and are replaced by and , respectively, with and nonnegative functions, singular or not). In this paper, Green’s functions associated with problem -, the inequalities satisfied by these functions, and the cone defined in the proof of the main results are different than the corresponding ones that the authors used in [10–12] for problem -. Existence results for the positive solutions of problem -, where and are sign-changing functions which may be singular at or and satisfy some different assumptions than those used in this paper, were obtained in [13]. We also mention the paper [14], where the authors studied the existence and multiplicity of positive solutions for system with , and the boundary conditions , , , and , with , , and and are sign-changing nonsingular or singular functions. The results obtained in [14] are relying on a nonlinear alternative of Leray-Schauder type and the Krasnosel’skii’s fixed point theorem. For other recent results concerning the coupled fractional boundary value problems we refer the reader to [15–17].

The paper is organized as follows. Section 2 contains some auxiliary results which investigate a nonlocal boundary value problem for fractional differential equations and presents the properties of Green’s functions associated to our problem -. In Section 3, we prove the main existence theorems for the positive solutions with respect to a cone for - which are based on the Guo-Krasnosel’skii fixed point theorem, and then the nonexistence of positive solutions is studied in Section 4. Finally, in Section 5, two examples are given to illustrate our main results.

2. Auxiliary Results

In this section, we present some auxiliary results that will be used to prove our main results.

We consider the fractional differential system

with the coupled integral boundary conditions

where , , , , and are functions of bounded variation.

Lemma 1 (see [13]). If are functions of bounded variation, and , then the pair of functions given by

is solution of problem (1)-(2), where

Lemma 2 (see [13]). The functions , given by (5) have the following properties:
(a) are continuous functions, and , for all .
(b) , for all , where and for all .
(c) , for all .

Lemma 3 (see [13]). If are nondecreasing functions, and , then given by (4) are continuous functions on and satisfy for all , (). Moreover, if satisfy , for all , then the solution of problem (1)-(2) given by (3) satisfies for all .

Lemma 4. Assume that are nondecreasing functions and . Then the functions satisfy for all the following relations:

, where .., where .., where .., where ..

Proof. The above inequalities follow from the properties of the functions , () from Lemma 2.

Lemma 5. Assume that are nondecreasing functions, , and , , and for all . Then the solution , of problem (1)-(2) given by (3) satisfies the inequalities , , for all .

Proof. By using Lemma 4, we obtain for all the following inequalities:

In the proof of our main existence results we will use the Guo-Krasnosel’skii fixed point theorem presented below (see [18]).

Theorem 6. Let be a Banach space and let be a cone in . Assume and are bounded open subsets of with and let be a completely continuous operator such that either

(i), , and , , or(ii), , and , .

Then has a fixed point in .

3. Existence Results for the Positive Solutions

In this section, we will give sufficient conditions on , and such that positive solutions with respect to a cone for our problem - exist.

We present now the assumptions that we will use in the sequel:

(H1) are nondecreasing functions and .(H2)The functions are continuous.

For , we introduce the following extreme limits:

In the definitions of the extreme limits above, the variables and are nonnegative.

By using Lemma 1, a solution of the following nonlinear system of integral equations

is a solution for problem -.

We consider the Banach space with the supremum norm and the Banach space with the norm . We define the cone by

For , we introduce the operators and defined by

and , . It is clear that if is a fixed point of operator , then is a positive solution of problem -.

Lemma 7. If - hold, then is a completely continuous operator.

Proof. Let be an arbitrary element. Because and satisfy problem (1)-(2) for , , and , , then by Lemma 5, we obtain

and so

By and the above inequalities, we deduce that . Hence, we get . By using standard arguments, we can easily show that and are completely continuous, and then is a completely continuous operator.

For , we denote , , , , , , , and , where , are defined in Section 2 (Lemma 4).

For and numbers , , , and , we define the numbers

where .

Theorem 8. Assume that and hold, , , , and and .
(1) If , , and , then, for each and , there exists a positive solution , for -.
(2) If , and , then, for each and , there exists a positive solution , for -.
(3) If , , and , then, for each and , there exists a positive solution , for -.
(4) If , , then, for each and , there exists a positive solution , for -.
(5) If or or , then, for each and , there exists a positive solution , for -.
(6) If or or , then, for each and , there exists a positive solution , for -.
(7) If or , or , then, for each and , there exists a positive solution , for -.
(8) If or or , then, for each and , there exists a positive solution , for -.

Proof. We consider the above cone and the operators , , and . Because the proofs of the above cases are similar, in what follows we will prove one of them, namely, Case . So, we suppose and . Let and . We choose and . Let be a positive number such that , and

By using and the definitions of and , we deduce that there exists such that and for all , with . We define the set . Now let , that is, with or equivalently . Then for all , and by Lemma 4, we obtain

Therefore, .
In a similar manner, we conclude

Hence, .
Then, for , we deduce

By the definitions of and , there exists such that and for all with and . We consider and we define . Then for with , we obtain

Then, by Lemma 4, we conclude

So, .
In a similar manner, we deduce

So, .
Hence, for , we obtain

By using (17), (21), Lemma 7, and Theorem 6(i), we conclude that has a fixed point such that , , and for all . If then for all , and if then for all .

In what follows, for and numbers , , , and , we define the numbers

Theorem 9. Assume that and hold, , , , , and .
(1) If and and , then, for each and , there exists a positive solution , for -.
(2) If , , and , then, for each and , there exists a positive solution , for -.
(3) If , , and , then, for each and , there exists a positive solution , for -.
(4) If , , then for each and , there exists a positive solution , for -.
(5) If or or , then, for each and , there exists a positive solution , for -.
(6) If or or , then, for each and , there exists a positive solution , for -.
(7) If or or , then, for each and , there exists a positive solution , for -.
(8) If or or , then, for each and , there exists a positive solution , for -.

Proof. We consider again the above cone and the operators , , and . Because the proofs of the above cases are similar, in what follows we will prove one of them, namely, the first case of . So, we suppose , , and . Let and . We choose and , and let be a positive number such that , and

By using and the definitions of and , we deduce that there exists such that and for all with and . We denote . Let with , that is, . Because for all , then, by using Lemma 4, we obtain

Therefore, .
Thus, for an arbitrary element , we deduce

Now, we define the functions , , , , . Then , for all , , , and . The functions are nondecreasing for every , and they satisfy the conditions

Therefore, for , there exists such that, for all and , we have

and so and .
We consider and we denote . Let . By the definitions of and , we obtain

Then, for all , we conclude

Therefore, .
In a similar manner, we deduce

So, .
Then, for , it follows that

By using (25), (31), Lemma 7, and Theorem 6(ii), we conclude that has a fixed point such that .

4. Nonexistence Results for the Positive Solutions

We present in this section intervals for and for which there exists no positive solution of problem - that can be viewed as fixed point of operator .

Theorem 10. Assume that and hold, and . If , then there exist positive constants , such that, for every and , the boundary value problem - has no positive solution.

Proof. In a similar manner as in the proof of Theorem from [11], we can show that and , where , , , and , satisfy the conditions of our theorem.

Theorem 11. Assume that and hold, and . If and for all , , and , then there exists a positive constant such that, for every and , the boundary value problem - has no positive solution.

Proof. From the assumptions of the theorem, we deduce that there exists such that for all and . We define , where and . We will show that, for every and , problem - has no positive solution.
Let and . We suppose that - has a positive solution .
If , then , and therefore, we obtain

Then we conclude

and so, , which is a contradiction.
If , then , and therefore, we deduce

Then we conclude

and so, , which is a contradiction.
Therefore, the boundary value problem - has no positive solution.

Theorem 12. Assume that and hold, and . If and for all , , and , then there exists a positive constant such that, for every and , the boundary value problem - has no positive solution.

Proof. From the assumptions of the theorem, we deduce that there exists such that for all and . We define , where and . Using a similar approach as that used in the proof of Theorem 11, we can show that, for every and , problem - has no positive solution.

Theorem 13. Assume that and hold, and . If and for all , , and , then there exist positive constants and such that, for every and , the boundary value problem - has no positive solution.

Proof. From the assumptions of the theorem, we deduce that there exist such that and , for all and .
We define and , where and . Then, for every and , problem - has no positive solution. Indeed, let and . We suppose that - has a positive solution . In a similar manner as that used in the proofs of Theorems 11 and 12, we obtain

and so

which is a contradiction. Therefore, the boundary value problem - has no positive solution.
We can also define and , where and . Then, for every and , problem - has no positive solution. Indeed, let and . We suppose that - has a positive solution . In a similar manner as that used in the proofs of Theorems 11 and 12, we obtain

and so

which is a contradiction. Therefore, the boundary value problem - has no positive solution.

5. Examples

Let (), (), , , for all . Then and .

We consider the system of fractional differential equations

with the boundary conditions

Then we obtain . The functions and are nondecreasing, and so assumption is satisfied. Besides, we deduce

We also obtain

For , we deduce . After some computations, we conclude , , , , , , , and .

Example 1. We consider the functions

for , , where and .
We have , , , and . We take ; then we obtain , , , and . The conditions and become and . If and , then the above conditions are satisfied. Therefore, by Theorem 8, for each and , there exists a positive solution , for problem -. For example, if , , , and , then we obtain , , , and .
Besides, because , , , and , we can apply Theorem 10. So, we conclude that there exist such that, for every and , the boundary value problem - has no positive solution. By Theorem 10, the positive constants and are given by and . For example, if , , , and , then we obtain , , , and .

Example 2. We consider the functions

where .
We have , , and . For , we obtain . Then, by Theorem 8, we conclude that, for each and , there exists a positive solution , for problem -.
Because and , we can apply Theorem 12. Then there exists such that, for every and , problem - has no positive solution. For example, if , then we deduce and .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper and regarding the funding that they have received.

Acknowledgments

The work of Rodica Luca and Alexandru Tudorache was supported by the CNCS Grant PN-II-ID-PCE-2011-3-0557, Romania.

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