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This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.

This paper considers the following boundary value problems of fractional order differential equations

where

Fractional order Differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Applications can be found in fields of control, porous media, eletromagnetic, etc. (see [

Recently some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator (

This paper generalizes the results of the papers above [

Some notions and Lemmas are important in order to state our results. Denote by

Definition 2.1 ( [

where

Definition 2.2 ( [

where

A solution of the problem (1.1) is defined as follows.

Definition 2.3 A function

Lemma 2.1 ( [

has solutions

Lemma 2.2 Let

In particular, when

for some

Proof. By (2.1), (2.2),

where

Lemma 2.3 ( [

is valid in following case

As a consequence of Lemmas 2.1, Lemmas 2.2 and Lemmas 2.3, the following result is useful in what follows.

Lemma 2.4 Let

if and only if

Proof. Assume

And the following simple calculation can be obtained by (2.4)

Hence Equation (2.5). Conversely, it is clear that if

In this section, Our first result is based on the Banach fixed point theorem (see [

Theorem 3.1 Assume that

(H1) There exists a function

If

Then the BVP (1.1) has a unique solution on J.

Proof. Transform the problem (1.1) into a fixed point problem. Consider the operator

defined by

The Banach contraction principle is used to prove that T has afixed point.

Let

Thus

Consequently, by (3.1) T is a contraction operator. As a consequence of the Banach Fixed point theorem, T has a fixed point which is the unique solution of the problem (1.1). The proof is completed.

In Theorem 3.1, if the function

Theorem 3.2 Assume that

(H2) There exists a constant L > 0 (i.e.

If

Then the BVP (1.1) has a unique solution on J.

The third result is based on Schauder’s Fixed point theorem.

Theorem 3.3 Assume that

(H3) The function

(H4) There exists a constant M > 0, such that

Then the BVP (1.1) has at least one solution on J.

Proof. Schauder’s Fixed point theorem is used to prove that T defined by (3.2) has a fixed point. The proof will be given in several steps.

Step 1: T is continuous.

Let

then

Since f is a continuous function, it can be shown that

And hence

Step 2: T maps the bounded sets into the bounded sets in

For any

In fact,

Thus

where

Step 3: Tmaps the bounded sets into the equicontinuous sets of

Let

Then

As

Step 4: A priori bounds.

Let

Let

By the condition (H4) and Step 2,

Thus for every

This shows that the set

In Theorem 3.3, if the condition (H4) is weakened, the fourth result can be obtained, which is a more general existence result (see [

Theorem 3.4 Assume that (H3) and the following conditionshold.

(H5) There exist a functional

(H6) There exists a number K > 0, such that

Then the BVP (1.1) has at least one solution on J.

Proof. Consider the operator T defined by (3.2),

By (H6), there exists K such that

Therefore, T is Leray-Schauder type operator (see [

For the boundary value problem

Take

Let

Hence the condition (H1) holds with

only if

For example,

Then by Theorem 3.1 the boundary value problem (4.1) has a uniquesolution on

The authors would like to thank the reviewers for their valuable suggestions and comments, which improved the completeness of the paper. Research of J. Song is funded by the High-level Talents Project of Guangdong Province Colleges and Universities (2013-178). Research of Z. Zeng is funded by the Natural Science Foundation of Guangdong Province of China (S2012010010069). These supports are greatly appreciated.

Sun, Y.F., Zeng, Z. and Song, J. (2017) Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation. Applied Mathematics, 8, 312-323. https://doi.org/10.4236/am.2017.83026