# Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces

- Zhi-Wei Lv
^{1}, - Jin Liang
^{2}Email author and - Ti-Jun Xiao
^{3}

**2010**:340349

**DOI: **10.1155/2010/340349

© Zhi-Wei Lv et al. 2010

**Received: **4 January 2010

**Accepted: **8 February 2010

**Published: **14 February 2010

## Abstract

A new existence and uniqueness theorem is given for solutions to differential equations involving the Caputo fractional derivative with nonlocal initial condition in Banach spaces. An application is also given.

## 1. Introduction

Fractional differential equations have played a significant role in physics, mechanics, chemistry, engineering, and so forth. In recent years, there are many papers dealing with the existence of solutions to various fractional differential equations; see, for example, [1–6].

In this paper, we discuss the existence of solutions to the nonlocal Cauchy problem for the following fractional differential equations in a Banach space :

where is the standard Caputo's derivative of order , and is a given -valued function.

## 2. Basic Lemmas

Let be a real Banach space, and the zero element of . Denote by the Banach space of all continuous functions with norm . Let be the Banach space of measurable functions which are Lebesgue integrable, equipped with the norm . Let , function is called a solution of (1.1) if it satisfies (1.1).

Recall the following defenition

Definition 2.1.

*Kuratowski measure of noncompactness*of is defined as

Clearly, . For details on properties of the measure, the reader is referred to [2].

where is the gamma function.

Remark.

Caputo's derivative of a constant is equal to .

Lemma (see [7]).

Lemma (see [7]).

Lemma(see [9]).

If is bounded and equicontinuous, then

(a) ;

(b) where

Lemma.

where , .

Proof.

## 3. Main Results

, and there exist such that for and each

Lemma.

Proof.

Conversely, if is a solution of (3.2), then for every , according to Remark 2.4 and Lemma 2.5, we have

It is obvious that This completes the proof.

Theorem.

If (H_{1}) and (H_{2}) hold, then the initial value problem (1.1) has at least one solution.

Proof.

Clearly, the fixed points of the operator are solutions of problem (1.1).

It is obvious that is closed, bounded, and convex.

Step 1.

We prove that is continuous.

Let

Step 2.

We prove that .

Let . Then for each , we have

Step 3.

We prove that is equicontinuous.

Let , and . We deduce that

As , the right-hand side of the above inequality tends to zero.

Step 4.

We prove that is relatively compact.

Let be arbitrarily given. Using the formula

From (3.17), we see that is relatively compact. Hence, is completely continuous. Finally, the Schauder fixed point theorem guarantees that has a fixed point in .

Theorem.

Then, the solution of (1.1) is unique in .

Proof.

By (3.18), we obtain So, the two solutions are identical in .

## 4. Example

Let

with the norm Consider the following nonlocal Cauchy problem for the following fractional differential equation in :

Conclusion 4.

Problem (4.2) has only one solution on

Proof.

So, is satisfied.

In the same way as in Example in [9], we can prove that is relatively compact in .

By a direct computation, we get

Hence, condition is also satisfied.

Moreover, we have

Therefore, . Thus, our conclusion follows from Theorem 3.3.

## Declarations

### Acknowledgments

This work was supported partially by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

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