Positive Solutions for a Coupled System of Nonlinear Semipositone Fractional Boundary Value Problems

Abstract

In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation , , , where the coefficients are real positive constants, , are the standard Riemann-Liouville derivatives. Values of the parameters and are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.

1. Introduction

In recent years, fractional-order calculus has been one of the most rapidly developing areas of mathematical analysis. In fact, a natural phenomenon may depend not only on the time instant but also on the previous time history, which can be successfully modeled by fractional calculus. Fractional-order differential equations are naturally related to systems with memory, as fractional derivatives are usually nonlocal operators. Thus, fractional differential equations (FDEs) play an important role because of their applications in various fields of science, such as mathematics, physics, chemistry, optimal control theory, finance, biology, and engineering [1–6]. In particular, a great interest has been shown by many authors in the subject of fractional-order boundary value problems (BVPs), and a variety of results for BVPs equipped with different kinds of boundary conditions have been obtained; for instance, see [7–18] and the references cited therein.

We consider the four-point coupled system of nonlinear fractional differential equations:with the coupled boundary conditionswhere , and are the standard Riemann-Liouville derivatives, and are real positive constants.

Here we emphasize that our problem is new in the sense of nonseparated coupled boundary conditions introduced here. To the best of our knowledge, fractional-order coupled system (1) has yet to be studied with the boundary conditions (2). In consequence, our findings of the present work will be a useful contribution to the existing literature on the topic. The existence of positive solution results for the given problem is new, though they are proved by applying the well-known fixed point theorem.

We present intervals for parameters , and such that the above problem (1)-(2) has at least one positive solution. By a positive solution (1)-(2), we mean a pair of functions satisfying (1) and (2) with for all and

We use the following notations for our convenience:

Before stating our results, we make precise assumptions throughout the paper:(H1)The functions and there exist functions such that and for any and (H2) are positive constants such that (H3) for all .(H4)The functions , may be singular at andor , and there exist functions , such that for all , with (H5)There exists such that

The rest of the paper is organized as follows. In Section 2, we construct the Green functions for the associated linear fractional-order boundary value problems and estimate the bounds for these Green functions. In Section 3, we establish the existence of at least one positive solution of the boundary value problem (1)-(2) by applying fixed point theorem. Finally, as an application, we give an example to illustrate our result.

2. Green Functions and Bounds

In this section, we construct the Green functions for the associated linear fractional-order boundary value problems and estimate the bounds for these Green functions, which are needed to establish the main results.

Lemma 1. Let Then, the differential equation has a solution for some , where is the smallest integer greater than or equal to .

Lemma 2. Let Then, for some , where is the smallest integer greater than or equal to .

Lemma 3. Let and . Let be given functions. Then, the boundary value problem,has an integral representationwhere

Lemma 4. Assume that condition is satisfied. Then, the Green functions and defined, respectively, by (8) and (10) are nonnegative, for all

Lemma 5. Assume that condition is satisfied. Then, the Green functions and defined, respectively, by (9) and (11) are nonnegative, for all

Lemma 6. Assume that condition is satified. Then, the Green functions and defined, respectively, by (8) and (10) have the following properties: , for all ,, , for all , where

Lemma 7. Assume that condition is satified. Then, the Green functions and defined, respectively, by (9) and (11) have the following properties: and for all , and , for all , where

In the proof of our main results, we shall use the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel’skii fixed point theorem presented below [19, 20].

Theorem 8. Let be a Banach space with closed and convex. Assume is a relatively open subset of with , and let be a completely continuous operator (continuous and compact). Then, either (i) has a fixed point in , or(ii)there exist and such that .

Theorem 9 (). Let be a Banach space, and let be a cone in . Assume that and are two 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. Main Results

In this section, we investigate the existence of positive solutions for our problem (1)-(2).

We consider the system of nonlinear fractional differential equationswith the boundary conditionswhere a modified function for any by Here withis solution of the system of fractional differential equationswith the boundary conditionsUnder the assumptions and or and , we have for all

We shall prove that there exists a solution for the boundary value problem (12)-(13) with and on on In this case, with and represents a positive solution of boundary value problem (1)-(2).

By using Lemma 3, a solution of the systemis a solution for problem (12)-(13).

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

For , we define the operators and defined by with It is clear that if is a fixed point of operator , then is a solution of problem (12)-(13).

Lemma 10. If and or and hold, then operator is a completely continuous operator.

Proof. The operators and are well defined. To prove this, let be fixed with Then we have If and hold, then we deduce easily that and for all If and hold, we deduce, for all :where ,
Thus, is well defined.
Next, we show that For any fixed , by Lemmas 6 and 7, we have Similarly, Therefore, Hence, This implies that According to the Ascoli-Arzela theorem, we can easily get that is completely continuous.

Theorem 11. Assume that hold. Then, there exist constants and such that, for any and , the boundary value problem (1)-(2) has at least one positive solution.

Proof. Let be fixed. From and , there exist such thatWe define We will show that, for any and , problem (12)-(13) has at least one positive solution.
So, let and be arbitrary but fixed for the moment. We define the set We suppose that there exist or and such that or .
We deduce that Then by Lemma 3, for all , we obtainHence, and Then, , which is contradiction.
Therefore, by Theorem 8 (with ), we deduce that has a fixed point That is, or , and with and for all Moreover, by (25), we concludeTherefore, for all , and for all Let and for all Then, for all for all Therefore, is a positive solution of (1)-(2).

Theorem 12. Assume that , and hold. Then, there exist and such that, for any and , the boundary value problem (1)-(2) has at least one positive solution.

Proof. We choose a positive number and we define the set
We introducewith Let and Then, for any and , we haveThen, for any , we obtain Therefore,On the other hand, we choose a constant such that From , we deduce that there exists a constant such thatNow we define and let
We suppose that , that is, for all and Then, for any , we have or We deduce that or .
We suppose that . Then, for any , we obtain Therefore, we conclude Hence,Then, for any and , by (37) and (41), we deduce It follows that, for any , we obtain Then, andIf , then by a similar approach, we obtain again relation (44).
We suppose that , that is, , for all and Then, for any , we have Hence, or .
If , then for any we deduce in a similar manner as above that for all and Hence, we obtain relation (44). If , then in a similar way as above, we deduce again relation (44). Therefore, by Theorem 9, relation (35), and (44), we conclude that has a fixed point

4. Example

In this section, we give an example to illustrating our result. Let

Consider the system of fractional differential equations,where We have for all , and then assumption is satisfied. Besides, assumption is also satisfied, because and for all . Let and Then , . In addition, We also obtain , , , , , , , and then , . We can apply Theorem 11. So we conclude that there exist such that, for every and , the boundary value problem (47) has at least one positive solution.

5. Conclusions

This paper studies the existence of positive solution of a four-point coupled system of nonlinear fractional differential equations. We give sufficient conditions on , and such that the system has at least one positive solution. The existence of positive solution is discussed by using Guo-Krasnosel’skii fixed point theorem. Also, an example which illustrates the obtained result is presented.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that no competing interests exist.