# Existence and Uniqueness of Mild Solution for Fractional Integrodifferential Equations

- Fang Li
^{1}Email author and - GastonM N'Guérékata
^{2}

**2010**:158789

**DOI: **10.1155/2010/158789

© Fang Li and Gaston M. N'Guérékata. 2010

**Received: **1 April 2010

**Accepted: **17 June 2010

**Published: **13 July 2010

## Abstract

We study the existence and uniqueness of mild solution of a class of nonlinear fractional integrodifferential equations , , , in a Banach space , where . New results are obtained by fixed point theorem. An application of the abstract results is also given.

## 1. Introduction

An integrodifferential equation is an equation which involves both integrals and derivatives of an unknown function. It arises in many fields like electronic, fluid dynamics, biological models, and chemical kinetics. A well-known example is the equations of basic electric circuit analysis. In recent years, the theory of various integrodifferential equations in Banach spaces has been studied deeply due to their important values in sciences and technologies, and many significant results have been established (see, e.g., [1–11] and references therein).

On the other hand, many phenomena in Engineering, Physics, Economy, Chemistry, Aerodynamics, and Electrodynamics of complex medium can be modeled by fractional differential equations. During the past decades, such problem attracted many researchers (see [1, 12–21] and references therein).

where , and the fractional derivative is understood in the Caputo sense.

In this paper, motivated by [1–27] (especially the estimating approaches given in [4, 6, 10, 24, 27]), we investigate the existence and uniqueness of mild solution of (1.1) in a Banach space : generates a compact semigroup of uniformly bounded linear operators on a Banach space . The function is real valued and locally integrable on , and the nonlinear maps and are defined on into . New existence and uniqueness results are given. An example is given to show an application of the abstract results.

## 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 semigroup of uniformly bounded linear operators. Then there exists such that for .

According to [22, 23], a mild solution of (1.1) can be defined as follows.

Definition 2.1.

Remark 2.2.

In this paper, we use to denote the norm of whenever for some with . denotes the Banach space of all continuous functions endowed with the sup-norm given by for . Set .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnosel'skii).

- (i)
whenever ,

- (ii)
is compact and continuous,

- (iii)
is a contraction mapping.

Then there exists such that .

## 3. Main Results

We will require the following assumptions.

Theorem 3.1.

then (1.1) has a unique mild solution for every .

Proof.

Set , .

for . Hence .

The conclusion follows by the contraction mapping principle.

We assume the following.

(H3) The function is continuous, and there exists a positive function ( ) such that

and set

Let be the infinitesimal generator of a compact semigroup of uniformly bounded linear operators. Then there exists a constant such that for .

Theorem 3.2.

then (1.1) has a mild solution for every .

Proof.

where .

Let be the closed convex and nonempty subset of the space .

Set .

Thus, .

So, we know that is a contraction mapping.

Set .

which implies that is relatively compact in .

Next, we prove that is equicontinuous.

The right-hand side of (3.24) tends to as as a consequence of the continuity of in the uniform operator topology for by the compactness of . So as . Thus, , as , which is independent of . Therefore is compact by the Arzela-Ascoli theorem.

Next we show that is continuous.

So is continuous.

By Krasnosel'skii's theorem, we have the conclusion of the theorem.

Remark 3.3.

In Theorem 3.2, if we furthermore suppose that the hypothesis

holds, then we can obtain the uniqueness of the mild solution in Theorem 3.2.

which implies by Gronwall's inequality that (1.1) has a unique mild solution .

Example 3.4.

Then generates a compact, analytic semigroup of uniformly bounded linear operators.

, and .

then (1.1) has a unique mild solution by Theorem 3.2 and Remark 3.3.

## Declarations

### Acknowledgments

The authors are grateful to the referees for their valuable suggestions. The first author is supported by the NSF of Yunnan Province (2009ZC054M).

## Authors’ Affiliations

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