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^{1,2}

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We discuss the initial value problem for the nonlinear fractional differential equation

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order, so fractional differential equations have wider application. Fractional differential equations have gained considerable importance; it can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic.

In the recent years, there has been a significant development in fractional calculus and fractional differential equations; see Kilbas et al. [

With the deep study, many papers that studied the fractional equations contained more than one fractional differential operator; see [

Babakhani and Daftardar-Gejji in [

Zhang in [

In above two equations,

McRae in [

In this paper, we use similar method as in [

Since

In Section

In this section, we need the following definitions and lemmas that will be useful to our main results. These materials can be found in the recent literatures; see [

Let

Let

Letting

Next, we will list the following lemma from [

Let

Let

Let

Since

Let

Let

Let

Suppose that

Assume that the conditions of Lemma

Let

The function

In this section, we establish the existence and uniqueness criteria of solutions for initial value problem (

Assume that

Let

Let

For the need of proof, we define function

Clearly, from the definition of function

For any

Hence, we get

Now, we will give the existence of maximal and minimal solutions of initial value problem (

Let

For any

To prove (i), let

Similarly, we can get

To prove (ii), let

Therefore, we can define the sequences

Clearly, the sequences

In the following, we will prove that

We assume that for some

Suppose that the conditions of Theorem

We have proved

In this paper, we will present an example to illustrate the main results.

Consider the initial value problem of fractional differential equation

Choose

Since

Next, we will prove the existence of maximal and minimal solutions for initial value problem (

Let

In addition,

In this paper, we considered the initial value problem of nonlinear fractional differential equation

This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), supported by Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003), and supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).