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

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.


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][2][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][5][6][7][8][9][10][11][12][13][14][15][16][17][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][20][21][22][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 3 <  < 4,  : [0,1] ×  → , and    0+ 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  : 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.

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][2][3] and the references therein.
One can easily obtain the following property from the definition of Caputo derivative.
Finally, we give Schaefer's fixed point theorem.

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.
Our main result, based on Schaefer's fixed point theorem, is stated as follows.
Then, BVP (1) has at least one solution, provided that Proof.We divide the proof into the following two steps.
Step 2. A priori bounds.Set Now it remains to show that the set Ω is bounded.
In view of (15), we can see that there exists a constant  > 0 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.
Then one can see that