- Research Article
- Open Access

# Solvability of Nonautonomous Fractional Integrodifferential Equations with Infinite Delay

- Fang Li
^{1}Email author

**Received:**4 September 2010**Accepted:**29 October 2010**Published:**31 October 2010

## Abstract

We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with infinite delay in a Banach space . The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii's fixed point theorem. An application of the abstract results is also given.

## Keywords

- Banach Space
- Probability Density Function
- Mild Solution
- Fractional Differential Equation
- Dominate Convergence Theorem

## 1. Introduction

, defined by for , belongs to the phase space , and . The fractional derivative is understood here in the Riemann-Liouville sense.

In recent years, the fractional differential equations have been proved to be good tools in the investigation of many phenomena in engineering, physics, economy, chemistry, aerodynamics, electrodynamics of complex medium and they have been studied by many researchers (cf., e.g., [13, 14, 16, 17] and references therein). Moreover, many phenomena cannot be described through classical differential equations but the integral and integrodifferential equations in abstract spaces in fields like electronic, fluid dynamics, biological models, and chemical kinetics. So many significant works on this topic have been appeared (cf., e.g., [10, 15, 18–25] and references therein).

In this paper, we study the existence of mild solution of (1.1) and obtain the existence theorem based on the measures of noncompactness without the assumptions that the nonlinearity satisfies a Lipschitz type condition and the semigroup generated by is compact (see Theorem 3.1). An example is given to show an application of the abstract results.

## 2. Preliminaries

Next, we recall the definition of the Riemann-Liouville integral.

Definition 2.1 (see [26]).

where is the Gamma function. Moreover, , for all .

- (1)
(see [26]),

- (2)

where is a beta function.

See the following definition about phase space according to Hale and Kato [27].

Definition 2.3.

A linear space consisting of functions from into , with seminorm , is called an admissible phase space if has the following properties.

- (1)
- (2)
for and as in (1).

- (3)
The space is complete.

Remark 2.4.

Equation (2.4) in (1) is equivalent to , for all .

Next, we consider the properties of Kuratowski's measure of noncompactness.

Definition 2.5.

From the definition we can get some properties of -measure immediately, see [28].

Lemma 2.6 (see [28]).

Let and be bounded subsets of , then

- (1)
, if ;

- (2)
, where denotes the closure of ;

- (3)
if and only if is precompact;

- (4)
, ;

- (5)
;

- (6)
, where ;

- (7)
, for any .

where .

The following lemma will be needed.

Lemma 2.7.

If is a bounded, equicontinuous set, then

(i) ,

(ii) , for .

For a proof refer to [28].

Lemma 2.8 (see [29]).

We need to use the following Sadovskii's fixed point theorem here, see [30].

Definition 2.9.

Let be an operator in Banach space . If is continuous and takes bounded sets into bounded sets, and for every bounded set of with , then is said to be a condensing operator on .

Lemma 2.10 (Sadovskii's fixed point theorem [30]).

Let be a condensing operator on Banach space . If for a convex, closed, and bounded set of , then has a fixed point in .

In this paper, we denote that is a positive constant, and assume that a family of closed linear operators satisfying the following.

(A1)The domain of is dense in the Banach space and independent of .

where , , (see [31]).

According to [16], a mild solution of (1.1) can be defined as follows.

Definition 2.11.

To our purpose the following conclusions will be needed. For the proofs refer to [16].

Lemma 2.12 (see [16]).

## 3. Main Results

We need the hypotheses as follows:

and set ,

where , and ,

where , and .

Theorem 3.1.

Suppose that (H1)–(H3) are satisfied, and if , then for (1.1) there exists a mild solution on .

Proof.

It is easy to see that is well-defined.

Let , .

and , .

Thus is a Banach space.

Obviously, the operator has a fixed point if and only if has a fixed point. So it turns out to prove that has a fixed point.

This means that is continuous.

where .

where .

This means .

where .

This contradicts (3.3). Hence for some positive number , .

It follows from Lemma 2.12, (H1) and (3.23) that , , as .

So, the set is equicontinuous.

In view of the Sadovskii's fixed point theorem, we conclude that has at least one fixed point in . Let , , then is a fixed point of the operator which is a mild solution of (1.1).

Now we assume that

Theorem 3.2.

Assume that (H1') and (H2') are satisfied, then (1.1) has a unique mild solution.

Proof.

and the result follows from the contraction mapping principle.

Example 3.3.

, are continuous functions, and .

Then generates an analytic semigroup satisfying assumptions (A1)–(A3) (see [33]).

then we can see that in (2.5).

now .

Then the above equation (3.37) can be written in the abstract form as (1.1).

where , satisfy (H1).

Suppose further that

- (1)
there exists such that ,

- (2)
,

then (3.37) has a mild solution by Theorem 3.1.

## Declarations

### Acknowledgments

The author is grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (no. 2009ZC054M).

## Authors’ Affiliations

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