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By using the Banach fixed point theorem and step method, we study the existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations. Meanwhile, by citing some counterexamples, it is pointed out that there exist a few defects in the proofs of the known results.

Recently, fractional differential equations are applied widely in various fields of science and engineering. Regarding applications of fractional differential equations, we refer to [

In this paper we consider the cauchy problem (

This paper is organized as follows: in Section

In this section, we introduce some basic definitions and notations about fractional calculus. Meanwhile, several known theorems are given, which are useful in this paper.

Let

Let

Assume that

Let

In the space

Let

Let

Let

In the space

Let

(Banach Fixed Point Theorem) Let (U,d) be a nonempty complete metric space, let

Let

It should be worthy noting that the conditions in Theorems

In this section, we will establish several useful lemmas. It should be pointed out that, in [

Let

Since

If

If

Hence we have

This completes the proof of Lemma

If

Furthermore, we have the following conclusion.

The fractional integration operator

Firstly we prove that if

It is easy to see that as

Similarly, we can prove that as

Now we prove the estimate. In fact

This completes the proof of Lemma

Let

The proof is similar to the proof of Lemma

Next we give the estimate. Because

If

If

Hence we have

This completes the proof of Lemma

Next, on the basis of above lemmas, we establish the results about the existence and uniqueness of solution for the cauchy-type problem (

Let

First we prove the existence of a unique solution

We rewrite the integral (

To apply Theorem

It follows from (

By Theorem

Next we consider the interval

We rewrite the integral equation (

To apply Theorem

Since

Next we consider the interval

Then there exists a unique solution

To complete the proof of Theorem

If

By (

Thus

By

This completes the proof of Theorem

Let

It should be pointed out that the conditions in Theorem

Let

In this section, by citing some counterexamples we would like to point out that, in [

Let one consider the function

From the above definition of

The next example illustrates that there also exists a problem in [

Consider the function

It is evident that

Setting

The following example is for [

Let one consider the function

The same problem exists in [

In a sense, our lemmas and main results have remedied these defects.

In this paper, we first get several useful lemmas, especially Lemmas

The authors would like to thank the anonymous referee for his/her remarks about the evaluation of the original version of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant no. 10701023 and no. 10971221), the Natural Science Foundation of Shanghai (no. 10ZR1400100), and Chinese Universities Scientific Fund (B08-1).