Positive Solutions to -Order Fractional Differential Equation with Parameter

Abstract

By employing the properties of the sum operators, we investigate the solutions of the -order fractional differential equations multiple point boundary value problem (in short BVP) with the boundary conditions contains a parameter. We not only obtain the existence and uniqueness of solutions about this BVP but also construct an iterative scheme to approximate the solution which is important for practical application. An example is given to demonstrate the validity of our main results.

1. Introduction

In this paper, we focus on the -order fractional differential equations of multiple point BVP:where is the -order fractional derivative, and are continuous, ,  , , is a parameter.

Boundary value problem is an important branch of differential equation. In recent years, as a mathematical model, BVP can be processed in many engineering and scientific models. Fractional differential equations BVP has become a hot issue; see the monographs of Lakshmikantham and Vatsala [1], Rudin [2], Samko et al. [3], Agarwal et al. [4, 5], and Webb and Zima [6]. Many excellent results have been reported; see [7–23]. In [24], El-Sayed and Bin-Taher study the following m-point BVP:By using the technical of Kolmogorov compactness criterion, they obtained the results of existence at least of one positive solution. In [25], Zhang studied the following -order BVP:The authors obtained the existence of positive solutions. However, many aspects of the theory to the solutions of the fractional differential equations BVP need to be explored, such as the solutions of high-order multipoint BVP.

Motivated by the work mentioned above, we focus on the existence and uniqueness of positive solutions for the nonlocal BVP (1) by the technical of a fixed point theorem of a sum operator. In [12], we study the fractional equations with two and three point boundary conditions, where . In [20], we study the fractional equations of with integral boundary conditions. The new features of this paper are as follows. Firstly, the order of the fractional differential equation is high order with , and the boundary condition is multipoint boundary condition. Compared with [12], the BVP is more general; secondly, the boundary conditions are dependent on a parameter which is different from paper [20], so the properties of Green function and the construct of the operator are differential. The Green function and the operator are the key to the two BVP.

The paper is organized as follows. In Section 2, we recall some definitions and lemmas. In Section 3, the existence of positive solutions to BVP (1) is obtained. Finally, in Section 4, an illustrative example is also presented.

2. Preliminaries

We first recall four classes operator.

The basic space used in this paper is the space . Set

Let or and be a real number with An operator is said to be(1)positive homogeneous if it satisfies(2)subhomogeneous if it satisfies(3)-concave if it satisfies(4) is said to be a mixed monotone operator if it satisfies for any .

Lemma 1 (see [26]). Let and .   is a mixed monotone operator; is an increasing subhomogeneous operator for any , and there exists a constant such that Assume that(i)for any ,(ii)there exists a constant such that and ; Then, the operator equation has a unique solution in and for any initial values , constructing successively sequences , , , we have and as

Lemma 2. Given , the unique solution ofiswhere ,

Proof. By using the properties of the Riemann-Liouville fractional derivative, we can easily get the conclusion, so we omit.

In the follow, we verify the properties of .

Lemma 3. For all , is continuous on and

Proof. Form the definition of , we can get is continuous. Now, we need only to prove that (12) is satisfied. For all , we haveHence is held; the proof is complete.

3. Main Results

In this section, we establish the existence and uniqueness of positive solutions results for the problems (1).

Theorem 4. Assume that and are continuous. A function is a solution of the BVP (1) if and only if it is a solution of the integral equation

Proof. The proof of this result is quite similar to that given earlier in Lemma 3 and omitted.

Define operators and as follows:

Theorem 5. Assume that(H₁), , and increasing in , decreasing in , and there exists a constant such that ;(H₂)for all , there exists a constant such that and . Then, for any , BVP (1) has a unique positive solution in , where , and for any , construct successively the sequenceswe have and as

Proof. According to Theorem 4 and the definition of operators and , it is straightforward to show that is the solution of BVP (1) if and only if solves the operator equation . We only need to investigate the solution of the above operator equation.
Now, we will show that and satisfy all the assumptions of Lemma 1. To begin with, we prove that is a mixed monotones operator. In fact, for , , with , , we know that and for all . According to and Lemma 3, we getsince , that is, . Furthermore, an argument similar to the one used in shows that is increasing. Next we prove that satisfies (8). For any and , fromwe have for ,  . Hence (8) holds. Also, for any , taking into consideration, we haveThus, for any ,   Hence the operator is subhomogeneous. Now, we are in the position to show that and . It follows from , and Lemma 3 that, for any ,wherewhereThanks to , and , we getSoand, in consequence, and Thus, for , inequality holds, and then we get . An argument similar to the one used in shows thatwherewhereObviously, and ; thus we are led to the conclusion that with , i.e., . Hence condition (i) of Lemma 1 is proved. It remains to show that condition (ii) of Lemma 1 is satisfied. For any ,  , from we haveThat is, for any , inequality holds. Applying Lemma 2, we can get the conclusion of Theorem 5.

As a special case of Theorem 5, we obtain the following results.

Corollary 6. Assume that is increasing in , decreasing in and ;for all ,  ,  , there exists a constant such that . Then, the problemhas a unique positive solution in , where and, for any , constructing successively the sequenceswe have and as

Proof. Letting in Theorem 5, by suitable modification to the proof of Theorem 5, we can prove the desired conclusion.

If is a mixed monotone operator and satisfies is an increasing -concave operator. We can also obtain the following result.

Theorem 7. Assume that, , and increasing in , decreasing in , and there exists a constant such that ;for all ,  ,  , there exists a constant such that and . Then, BVP (1) has a unique positive solution in , where and, for any , constructing successively the sequenceswe have and as

Proof. In the light of [26, Theorem 2.4], a similar argument used in Theorem 5. shows that the desire result. We omit it.

4. Example

In this section, we give one example to illustrate our results.

Example 1. Consider the BVP (1). Letting , , , , , and A simple calculation gives . It is not difficult to verify that and are satisfied. Taking , we haveThus, is satisfied. Moreover, for any , , , , we havewhere We conclude that condition is satisfied. Therefore, Theorem 5 ensures that the BVP (1) has a unique positive solution in with .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

This project is supported by Shandong Provincial Natural Science Foundation (Grant no. ZR2017LA002), Weifang Science and Technology Development Projects (Grant no. 2017GX025), and Doctoral Research Foundation of Weifang University (Grant no. 2017BS02).