Solvability of a Class of Higher-Order Fractional Two-Point Boundary Value Problem

Abstract

This paper investigates the solvability of a class of higher-order fractional two-point boundary value problem (BVP), and presents several new results. First, Green’s function of the considered BVP is obtained by using the property of Caputo derivative. Second, based on Schaefer’s fixed point theorem, the solvability of the considered BVP is studied, and a sufficient condition is presented for the existence of at least one solution. Finally, an illustrative example is given to support the obtained new results.

1. Introduction

Fractional differential equations (FDEs) have been of great interest recently. This is due to the development of the theory of fractional calculus itself as well as its applications [1–3]. Apart from diverse areas of mathematics, fractional differential equations arise in engineering, technology, biology, chemical process, and many other branches of science. The first issue for the theory of fractional differential equations is the existence of solutions to kinds of boundary value problems (BVPs), which has been studied recently by many scholars [4–18], and lots of excellent results have been obtained by means of fixed-point theorems, Leray-Schauder theory, upper and lower solutions technique, and so forth.

It should be pointed out that as an important branch of fractional differential equations, higher-order FDEs have been studied in a series of recent works [19–23]. In [19], higher-order fractional heat-type equations were investigated and some interesting properties on the solution to this type of equations were presented. Goodrich [21] considered another kind of higher-order fractional differential equations and presented some results on the existence of one positive solution.

In this paper, we study the following higher-order fractional two-point boundary value problem: where , , and is the Caputo derivative. To our best knowledge, there are fewer results on the existence of solutions to BVP (1). Firstly, we establish Green’s function for BVP (1) by using the property of Caputo derivative. Secondly, based on Schaefer’s fixed point theorem, we present a sufficient condition for the existence of at least one solution. Throughout this paper, we assume that the nonlinearity is continuous. Moreover, let with the norm . Then, is a Banach space.

The rest of this paper is structured as follows. Section 2 contains some preliminaries on the Caputo derivative. Section 3 investigates the existence of solutions to BVP (1) and presents the main results of this paper.

2. Preliminaries

In this section, we give some necessary preliminaries on the Caputo derivative, which will be used in the sequel. For details, please refer to [1–3] and the references therein.

Definition 1 (see [3]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on .

Definition 2 (see [3]). The Caputo fractional derivative of order of a continuous function is given by where , provided that the right side is pointwise defined on .

One can easily obtain the following property from the definition of Caputo derivative.

Proposition 3 (see [3]). Let . Assume that , . Then the following equality holds: for some , , where .

Finally, we give Schaefer’s fixed point theorem.

Lemma 4 (see [24]). Let be a Banach space, and is a completely continuous operator. If the set , is bounded, then has at least a fixed point.

3. Main Results

In this section, we first convert BVP (1) into an equivalent operator equation and then present some new results on the existence of solutions to BVP (1).

Firstly, we convert BVP (1) into an equivalent operator equation.

Lemma 5. Given , the unique solution of is where

Proof. Assume that satisfies (5). Then, from Proposition 3, we have
From the boundary value condition , one can see that Thus, we have which together with the boundary value condition yields that Therefore, the unique solution of (5) is
The proof is completed.

Lemma 6. is a solution to BVP (1), if and only if , where

Our main result, based on Schaefer’s fixed point theorem, is stated as follows.

Theorem 7. Let be continuous. Assume that (H) there exist nonnegative functions such that Then, BVP (1) has at least one solution, provided that

Proof. We divide the proof into the following two steps.
Step 1. is completely continuous.
Let be an open bounded subset. By the continuity of , we can get that is continuous and is bounded. Moreover, there exists a constant such that , , . Thus, in view of the Arzelá-Ascoli theorem, we need only to prove that is equicontinuous.
For , , we have which implies that is equicontinuous.
Step 2. A priori bounds.
Set
Now it remains to show that the set is bounded.
For , we get . Thus, form , we obtain that In view of (15), we can see that there exists a constant such that
As a consequence of Schaefer’s fixed point theorem, we deduce that has a fixed point which is the solution to BVP (1). The proof is completed.

Finally, we give an illustrative example to support our new results.

Example 8. Consider the following BVP:
A simple calculation shows that .
Then one can see that which implies that (15) holds. By Theorem 7, BVP (20) has at least one solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (G61174036), the Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education, and the Graduate Independent Innovation Foundation of Shandong University (yzc10064).