Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Abstract

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: ,  , , where , , , and . The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

1. Introduction

Fractional calculus has wide applications in many fields of science and engineering, for example, fluid flow, biosciences, rheology, electrical networks, chemical physics, control theory of dynamical systems, and optics and signal processing [1].

Recently, nonlinear fractional differential equations have been discussed under the following boundary conditions (BCs for short): (1)Integer derivative BCs: , , , , , , , , , , ; see papers [2–7], respectively. (2)Integer derivative and integral BCs: , , , ; see papers [8, 9], respectively. (3)Integer and fractional derivative BCs: , , , , , , , , , , , , , ;  see papers [10–16], respectively.(4)Integer derivative and fractional integral BCs: , , ; see papers [17, 18], respectively.

Besides, there are some other BCs involved in fractional differential equations, such as nonlinear BCs; refer to [19, 20].

Motivated greatly by the above-mentioned works, in this paper, we study the following boundary value problem (BVP for short) of nonlinear fractional differential equation with fractional integral BCs as well as integer and fractional derivativewhere and denote the standard Caputo fractional derivatives and denotes the standard Riemann-Liouville fractional integral. Throughout this paper, we always assume that , , , , and is continuous.

In order to prove our main results, the following well-known fixed point theorems are needed.

Theorem 1 (nonlinear alternative of Leray-Schauder type [21]). Let be a Banach space with closed and convex. Assume is a relatively open subset of with and is a continuous and compact map. Then either(a) has a fixed point in or(b)there exists and such that .

Theorem 2 (Banach contraction principle [22]). Let be a complete metric space and be contractive. Then has a unique fixed point in .

2. Preliminaries

In this section, we always assume that , , and denotes the integer part of . Now, for the convenience of the reader, we give the definitions of the Riemann-Liouville fractional integrals and fractional derivatives and the Caputo fractional derivatives on a finite interval of the real line, which may be found in [1].

Definition 3. The Riemann-Liouville fractional integrals and of order on are defined by respectively.

Definition 4. The Riemann-Liouville fractional derivatives and of order on are defined by respectively, where

Definition 5. Let and be the Riemann-Liouville fractional derivatives of order . Then the Caputo fractional derivatives and of order on are defined by respectively, where

Lemma 6 (see [23]). If , then the equation , , is satisfied for .

Lemma 7 (see [23]). Let . Then the equation , , is satisfied for .

Lemma 8 (see [1]). Let be given by (5). Then the following relations hold:(1)For , .(2)If , then .

Lemma 9 (see [1]). Let be given by (5) and . Then where , .

For any , we define

Lemma 10. Let be nonnegative. Then , .

Proof. For any , we have

3. Main Results

In the remainder of this paper, for any nonnegative function , we denote and for any , we use the norm

Lemma 11. Let be a given function. Then the BVPhas a unique solution where

Proof. It follows from the equation in (11) and Lemma 9 thatSo, In view of (14), (16), and the BCs , we get and so, Then, by using Lemmas 6, 7, and 8, we may obtainwhich together with the BC implies that Therefore, the BVP (11) has a unique solution

Lemma 12. Let be nonnegative. Then

Proof. In view of Lemma 10, we have

Now, we define an operator by Obviously, is a solution of the BVP (1) if and only if is a fixed point of .

Theorem 13. Assume that , , and there exist nonnegative functions , nonnegative increasing continuous function defined on , and such thatThen the BVP (1) has one nontrivial solution.

Proof. Let . Since and are continuous on and , respectively, we may denoteFirst, we prove that is continuous. Suppose that (), , and (). Then for any and , we have . This together with (27) and (28) implies that, for any and , By applying Lebesgue dominated convergence theorem, we get which indicates that is continuous.
Next, we show that is compact. Assume that is a subset of . Then for any , we haveIn what follows, we will prove that is relatively compact. On the one hand, for any , there exists such that , and so, it follows from (27), (28), and (31) that which shows that is uniformly bounded. On the other hand, for any , since is uniformly continuous on , there exists such that, for any with ,For any , there exists such that , and so, for any with , it follows from (28), (31), and (33) that which indicates that is equicontinuous. By Arzela-Ascoli theorem, we know that is relatively compact. Therefore, is compact.
Now, we will prove that (a) of Theorem 1 is fulfilled. Suppose on the contrary that (b) of Theorem 1 is satisfied; that is, there exists and such that . Then, in view of (25), (26), and Lemma 12, we have which shows that This contradicts the fact .
So, it follows from Theorem 1 that has a fixed point , which is a desired solution of the BVP (1). At the same time, since , we know that the zero function is not a solution of the BVP (1). Therefore, is a nontrivial solution of the BVP (1).

Theorem 14. Assume that there exists a nonnegative function such thatThen the BVP (1) has a unique solution.

Proof. For any , in view of (37) and Lemma 12, we have This indicates that which together with (38) implies that is contractive. So, it follows from Theorem 2 that has a unique fixed point, and so, the BVP (1) has a unique solution.

Example 15. We consider the BVP

Let , . Then is continuous and , .

If we choose , , , and , , then it is easy to verify that (25) is satisfied.

Since , , , and , a direct calculation shows that If we choose , then (26) is fulfilled.

Therefore, it follows from Theorem 13 that the BVP (41) has one nontrivial solution.

Example 16. We consider the BVP

Let , . Then is continuous.

If we choose , then we may assert that (37) is satisfied. In fact, for any , if , then (37) is obvious. When , we may suppose that . In this case, by Lagrange mean value theorem, there exists such that, for any , that is, (37) is satisfied.

On the other hand, in view of , we know that (38) is fulfilled.

Therefore, it follows from Theorem 14 that the BVP (43) has a unique solution.

Competing Interests

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

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11661049).