^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems,

Once the models of fractional differential equation for the actual problem have been established, people immediately faced the problem of how to solve these models. In many cases, it is very difficult to obtain the exact solution of the fractional differential equation. So it requires researchers to find as many characteristics of the solution of the problem as possible. For example, does the equation have a solution? If there is one solution, is the solution unique? How can we compare the size of the solution? We noted that although there were many works with respect to fractional differential equations, which were shown in [

Al-Bassam [

Delbosco and Rodino [

In [

In [

We refer the readers to monographs [

The rest of the paper is organized as follows. In Section

For

The following relation holds

Supposing that

Suppose that

By the use of the continuity of

A function

A function

If one of the above inequalities is strict, then we call it as a strict lower (upper) solution.

Clearly, if functions

Many methods can be applied to study the existence of solution. However, generally speaking, it is nothing more than two ways. One is based on the method of the approximate solution of exact solution to prove the existence of the solution, namely, classical successive approximation method. A. Cauchy, R. Lipschitz, G. Peano, and so forth used this method to solve the existence of some special types of differential equations. In 1893, C. Picard applied this method to study the general nonlinear differential equation and established the existence and uniqueness of solutions, named the Cauchy-Picard Theorem. This method itself also contains a structural method to obtain the exact solution and thus provides a way for the approximate solution. Another method is transforming the solution into the fixed point of some maps. Although the method cannot give the approximate solution, it is the abstraction and generalization of the former method and is simple to use. In this section, we will establish the uniqueness of the solution for fractional IVP (

Assume that

For

Clearly, the operator

Now we prove that operator

Taking into account that the function

Similar to paper [

The study about the following problem is meaningful:

Firstly, let us discuss the result about the strict inequalities for fractional IVP.

Assume that the functions

Without loss of generality, suppose that

Similarly, let

Suppose for contradiction that conclusion (

Taking into account that

A standard proof can show that

The following conclusion is about the nonstrict inequalities.

Assume that the functions

Given

From (

For

The rest of the proof is just similar to Theorem

If we instead use condition (

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

This work is supported by NNSF (61174078, 61201431), SDNSF (ZR2010AM035), a project of Shandong Higher Educational Science and Technology program (J11LA07), the Taishan Scholar project, Research Award Fund for Outstanding Young Scientists of Shandong (BS2012SF022), and SDUST Research Fund (2011KYTD105). The authors thank the referee for his/her valuable comments and constructive suggestions.