# Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

- Fang Li
^{1}Email author

**2010**:287861

**DOI: **10.1155/2010/287861

© Fang Li. 2010

**Received: **8 January 2010

**Accepted: **21 January 2010

**Published: **26 January 2010

## Abstract

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . General existence and uniqueness theorem, which extends many previous results, are given.

## 1. Introduction

The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensively (see, e.g., [1–8] and references therein).

In this paper, we discuss the existence and uniqueness of mild solution for

where , , and generates an analytic compact semigroup of uniformly bounded linear operators on a Banach space . The term which may be interpreted as a control on the system is defined by

where (the set of all positive function continuous on ) and

The functions and are continuous.

The nonlocal condition can be applied in physics with better effect than that of the classical initial condition . There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 9–14] and references cited there).

In this paper, motivated by [1–7, 9–15] (especially the estimating approach given by Xiao and Liang [14]), we study the semilinear fractional differential equations with nonlocal condition (1.1) in a Banach space , assuming that the nonlinear map is defined on and is defined on where , for , the domain of the fractional power of . New and general existence and uniqueness theorem, which extends many previous results, are given.

## 2. Preliminaries

In this paper, we set , a compact interval in . We denote by a Banach space with norm . Let be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators , that is, there exists such that ; and without loss of generality, we assume that . So we can define the fractional power for , as a closed linear operator on its domain with inverse , and one has the following known result.

Lemma 2.1 (see [15]).

is a Banach space with the norm for .

for each and .

For every and , .

Definition 2.2.

for is called a mild solution of (1.1).

In this paper, we use to denote the norm of whenever for some with . We denote by the Banach space endowed with the sup norm given by

for .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnoselkii, see [16]).

Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that

whenever .

is compact and continuous,

is a contraction mapping.

Then there exists such that .

## 3. Main Results

- (H1)The function is continuous, and there exists a positive function such that(3.1)
where .

- (H2)The function is continuous and there exists such that(3.2)

for any .

Theorem 3.1.

Let be the infinitesimal generator of an analytic compact semigroup with and . If the maps and satisfy (H1), (H2), respectively, and , then (1.1) has a mild solution for every .

Proof.

where .

Let .

Hence we obtain .

which means that is continuous.

we can see that is uniformly bounded on .

It follows from the assumption of that tends to 0 as . For , using the Hölder inequality, we can see that tends to 0 as and .

Using the compactness of in implies the continuity of for integrating with , we see that tends to , as . For , from the assumption of and the Hölder inequality, it is easy to see that tends to 0 as and .

Thus, , as , which does not depend on .

So, is relatively compact. By the Arzela-Ascoli Theorem, is compact.

We now conclude the result of the theorem by Krasnoselkii's theorem.

Now we assume the following.

(H3) There exists a positive function such that

the function belongs and

Theorem 3.2.

Let be the infinitesimal generator of an analytic semigroup with and . If and (H2)–(H4) hold, then (1.1) has a unique mild solution .

Proof.

Obviously, is well defined on .

and the result follows from the contraction mapping principle.

## Declarations

### Acknowledgment

This work is supported by the NSF of Yunnan Province (2009ZC054M).

## Authors’ Affiliations

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