#### Abstract

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.

#### 1. Introduction

Recently, fractional differential equations are applied widely in various fields of science and engineering. Regarding applications of fractional differential equations, we refer to [1–15] and references cited therein. However, the investigation of basic theory of fractional differential equations is still not complete, and there is a great deal of work which needs to be done. Most of the investigations in this field involve the existence and uniqueness of solutions to fractional differential equations on the finite interval . In 1938, Pitcher and Sewell[16] first considered the nonlinear fractional differential equation

with the following initial conditions:

where , and is Riemann-Liouville fractional derivative. Barrett [17], in 1954, first considered the Cauchy-type problem for the linear fractional differential equation

with the same initial conditions (1.2). Afterwards, there is a great deal of work about the basic theory [18–27]. In [28], Kilbas et al. summarized systematically the main results.

In this paper we consider the cauchy problem (1.1)-(1.2); here can be Riemann-Liouville fractional derivative, and Hadamard-type fractional derivative. We establish some results about the existence and uniqueness of solution of (1.1)-(1.2). By the way, we will point out that there exist several defects in the proofs of the related theorems of [28].

This paper is organized as follows: in Section 2, we introduce some preliminaries and notations; main results are proved in Section 3; in Section 4, by citing several counterexamples, we will point out the defects in [28]; Section 5 is a brief summary of this paper.

#### 2. Preliminaries and Notations

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.

*Definition 2.1 (see [28]). *Let be a finite interval on the real axis . The Riemann-Liouville left-sided fractional integral of the function with order is defined by
where the real function is defined on the interval and the right-side integral of the above equality is assumed to make sense.

*Definition 2.2 (see [28]). *Let be a finite interval on the real axis . The Riemann-Liouville left-sided fractional derivative of the function with order is defined by
where the real function is defined on the interval and the right side of the above equality is assumed to make sense.

*Definition 2.3. *Assume that is defined on the set . is said to satisfy Lipschitzian condition with respect to the second variable, if for all and for any one has
where does not depend on .

*Definition 2.4 (see [28]). *Let , then the space is defined by
Here is a weighted space of continuous functions
and is the Riemann-Liouville fractional derivative.

In the space we define the norm

*Definition 2.5 (see [28]). *Let be a finite or infinite interval of the half-axis The Hadamard type left-sided fractional integral of the function with order is defined by
where and the right-side integral of the above equality is assumed to make sense.

*Definition 2.6 (see [28]). *Let be the -derivative. The Hadamard left-sided fractional derivative of the function on () with order is defined by
where , , and the right side of the above equality is assumed to make sense.

*Definition 2.7 (see [28]). *Let and The space is defined by
where is a Hadamard left-sided fractional derivative, and is a weighted space of continuous functions

In the space , we define the norm

Theorem 2.8 (see [28]). *Let , . Let be a function such that for any . If , then satisfies the relations:
**
if and only if satisfies the Volterra integral equation
**
where
**
where is a Riemann-Liouville left-sided fractional derivative.*

Theorem 2.9 (see [28]). *(Banach Fixed Point Theorem) Let (U,d) be a nonempty complete metric space, let , and let be a map such that, for every , the relation
**
holds. Then the operator T has a unique fixed point *

Theorem 2.10 (see [28]). *Let , and Let be a function such that for any If , then satisfies
**
if and only if satisfies the Volterra integral equation
**
where is a Hadamard-type left-sided fractional derivative.*

*Remark 2.11. *It should be worthy noting that the conditions in Theorems 2.8 and 2.10 are a little different from the ones in [28, pages 163, 213]. In [28], is an open set in and is assumed to be a function such that for any . In fact, we think that such assumption is not complete for the proof of the related conclusion.

#### 3. Main Results

In this section, we will establish several useful lemmas. It should be pointed out that, in [28], some analogous lemmas play important roles in the proofs of the related results. However, we have found out that there exist a few defects in these lemmas of [28], which means that the proofs of the related results in [28] are not complete. Several counterexamples will be given in Section 4. In a sense, our lemmas are to mend these cracks. Furthermore, several theorems about the existence and uniqueness of solution for the cauchy-type problem (2.10)-(2.11) will be given; then, in the sense of Hadamard fractional derivative, we have the similar result.

Lemma 3.1. *Let and Then and
*

*Proof. *Since and , then and . Now we prove the estimate. Because , there exists such that

If , then

If , then

Hence we have

This completes the proof of Lemma 3.1.

Lemma 3.2 (see [28]). *If , then the fractional integration operator with order is a mapping from to , and
**
here is a Riemann-Liouville fractional integral operator and .*

Furthermore, we have the following conclusion.

Lemma 3.3. *The fractional integration operator with order is a mapping from to , and
**
where is a Riemann-Liouville fractional integral operator and *

*Proof. *Firstly we prove that if , then . For any and , we have

It is easy to see that as we have

Similarly, we can prove that as we have

Thus .

Now we prove the estimate. In fact

This completes the proof of Lemma 3.3.

Lemma 3.4. *Let , , and Then and
*

*Proof. *The proof is similar to the proof of Lemma 3.1. Since and , we have , that is,

Next we give the estimate. Because , there exists at least such that

If , then

If , then

Hence we have

This completes the proof of Lemma 3.4.

Next, on the basis of above lemmas, we establish the results about the existence and uniqueness of solution for the cauchy-type problem (2.10)-(2.11) in the sense of Riemann-Liouville fractional derivative and Hadamard fractional derivative.

Theorem 3.5. *Let and . Let be a function such that for any and the Lipschitzian condition holds with respect to the second variable . Then there exists a unique solution for the cauchy-type problem (2.10)-(2.11).*

*Proof. *First we prove the existence of a unique solution . According to Theorem 2.8, it is sufficient to prove the existence of a unique solution to the nonlinear Volterra integral equation (2.12). Equation (2.12) makes sense in any interval Choose such that
where is the Lipschitzian coefficient. Next we prove the existence of a unique solution to (2.12) on the interval . For this, we use the Banach fixed point theorem for the space , which is a complete metric space with the distance given by

We rewrite the integral (2.12) in the form

where

To apply Theorem 2.9, we have to prove the following: () if , then ; () for any the following estimate holds:

It follows from (2.13) that . Since for any , then, by Lemma 3.2 [28] (with and , the integral in the right-hand side of (3.19) also belongs to , and hence . Now we prove the estimate in (3.21). By (3.20), using the Lipschitzian condition and applying the relation (3.6) (with , and , we have

which yields the estimate (3.21). In accordance with (3.17), , and hence, by Theorem 2.9, there exists a unique solution to (2.12) on the interval .

By Theorem 2.9, this solution is a limit of a convergent sequence :

where is any function in . If there is at least one in the initial condition (2.11), then we can take with defined by (2.13). The last relation can be rewritten into the form
where

Next we consider the interval . Rewrite (2.12) in the form

where is defined by
We obtain . Next we prove the existence of a unique solution to (2.12) on the interval . For this, we also use Banach fixed point theorem for the space , where satisfies
is a complete metric space with the distance given by

We rewrite the integral equation (3.26) into the form

where

To apply Theorem 2.9, we have to prove the following: () if , then ; () for any , the following estimate holds:

Since for any , then, by Lemma 3.3, the integral in the right-hand side of (3.31) also belongs to , and hence . Now we prove the estimate in (3.32) as follows:

which yields the estimate (3.32). In accordance with (3.28), then , and hence by Theorem 2.9, there exists a unique solution to (2.12) on the interval . By Theorem 2.9, this solution in is a limit of a convergent sequence :
where is any function in . If on , then we can take with defined by (2.13). The last relation can be rewritten in the form
where

Next we consider the interval where such that and . Using the same arguments as the above, we derive that there exists a unique solution to (2.12) on the interval . If , then take the next interval , where and such that and . If , repeating the above process, then we find that there exists a unique solution to (2.12), and , where and , and we take and on each interval . By we know that by finite steps we can arrive at

Then there exists a unique solution to (2.12) on the interval . By Lemma 3.1, we obtain that there exists a unique solution to the Volterra integral equation (2.12) on the whole interval [], and hence is the unique solution to the cauchy-type problem (2.10)-(2.11).

To complete the proof of Theorem 3.5, we must show that such a unique solution belongs to the the space ; it is sufficient to prove that . By the above proof, the solution is a limit of the sequence , where :

with the choice of certain on each .

If , then we can take .

By (2.10) and the Lipschitzian-condition, we have

Thus

By and for any , we have , that is, Hence .

This completes the proof of Theorem 3.5.

Corollary 3.6. *Let and Let be a function such that for any ; the Lipschitzian condition holds with respect to and and exist for any . Then there exists a unique solution for the cauchy-type problem (2.10)-(2.11).*

*Remark 3.7. *It should be pointed out that the conditions in Theorem 3.5 are different from the ones in [28, Theorem , page 165]. In [28], is an open set in and is assumed to be a function such that for any . In fact, such assumptions are not complete for the proof of the related conclusion. A counterexample will be given in Section 4. There exists the similar problem in [28, Theorem , page 213]. By applying Lemma 3.4, modifying the conditions in [28, Theorem , page 213] and using the similar arguments to the proof of Theorem 3.5, we arrive at the following result.

Theorem 3.8. *Let , and such that Let be a function such that for any and the Lipschitzian condition holds with respect to . Then there exists a unique solution for the Cauchy-type problem
**
in the space where is a Hadamard fractional derivative.*

#### 4. Counterexamples

In this section, by citing some counterexamples we would like to point out that, in [28, Lemmas 3.4, 3.9, and 3.10, pages 165, 202, and 213] are not complete.

*Example 4.1. *Let one consider the function
where is a constant number and .

From the above definition of , we know that and , but we cannot get the conclusion and Hence the conclusion of [28, Lemma , page 165] does not hold. We cannot apply it to prove [28, Theorem , page 165]. Furthermore, there also exists a problem about in [28, Theorem , page 165]. For example, choosing , we know that for any , satisfies Lipschitz condition with respect to the second variable . However, choosing , we can not arrive at . Hence the condition of in [28, Theorem , page 165] is not proper.

The next example illustrates that there also exists a problem in [28, Lemma , page 202].

*Example 4.2. *Consider the function

It is evident that belongs to the space = , = .

Setting , that is, , then

We could not conclude that
because
However, we have

The following example is for [28, Lemma , page 213].

*Example 4.3. *Let one consider the function
where is a constant number and .

The same problem exists in [28, Lemma , page 213]. From the definition of , we have and . However, the conclusion that and is still not correct. This defect means that [28, Lemma ] could not be applied to prove [28, Theorem ].

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

#### 5. Conclusion

In this paper, we first get several useful lemmas, especially Lemmas 3.1 and 3.4, which have improved the corresponding lemmas in [28]. By modifying the conditions on and improving the method used in [28], we have established the results of existence and uniqueness of solution for the cauchy-type problems involving the Riemann-Liouville fractional derivative and the Hadamard fractional derivative in the weight space of continuous functions. Meanwhile, we have given some counterexamples to prove that [28, Lemmas 3.4, 3.9, and 3.10, pages 165, 203, and 213] are not complete, which means that there exist some defects in the proofs of the related results in [28].

#### Acknowledgments

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).