- Research Article
- Open Access

# Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay

- Fang Li
^{1}Email author and - Jun Zhang
^{2}

**2011**:963463

https://doi.org/10.1155/2011/963463

© Fang Li and Jun Zhang. 2011

**Received:**5 December 2010**Accepted:**30 January 2011**Published:**24 February 2011

## Abstract

We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space . An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski's measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodifferential equation.

## Keywords

- Banach Space
- Phase Space
- Probability Density Function
- Positive Function
- Mild Solution

## 1. Introduction

The fractional differential equations are valuable tools in the modeling of many phenomena in various fields of science and engineering; so, they attracted many researchers (cf., e.g., [1–6] and references therein). On the other hand, the integrodifferential equations arise in various applications such as viscoelasticity, heat equations, and many other physical phenomena (cf., e.g., [7–10] and references therein). Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (cf., e.g., [7, 10–15] and references therein).

Neutral functional differential equations arise in many areas of applied mathematics and for this reason, the study of this type of equations has received great attention in the last few years (cf., e.g., [12, 14–16] and references therein). In [12, 16], Hernández and Henríquez studied neutral functional differential equations with infinite delay. In the following, we will extend such results to fractional-order functional differential equations of neutral type with infinite delay. To the authors' knowledge, few papers can be found in the literature for the solvability of the fractional-order functional integrodifferential equations of neutral type with infinite delay.

where , , is a phase space that will be defined later (see Definition 2.5). is a generator of an analytic semigroup of uniformly bounded linear operators on . Then, there exists such that . , , ( ), and defined by , for , belongs to and . The fractional derivative is understood here in the Caputo sense.

The aim of our paper is to study the solvability of (1.1) and present the existence of mild solution of (1.1) based on Kuratowski's measures of noncompactness. Moreover, an example is presented to show an application of the abstract results.

## 2. Preliminaries

Throughout this paper, we set and denote by a real Banach space, by the Banach space of all linear and bounded operators on , and by the Banach space of all -valued continuous functions on with the uniform norm topology.

Let us recall the definition of Kuratowski's measure of noncompactness.

Definition 2.1.

This measure of noncompactness satisfies some important properties.

Lemma 2.2 (see [17]).

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 ,

- (8)
, where is the closed convex hull of .

where .

The following lemmas will be needed.

Lemma 2.3 (see [17]).

Lemma 2.4 (see [18]).

The following definition about the phase space is due to Hale and Kato [11].

Definition 2.5.

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

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

- (3)
The space is complete.

Remark 2.6.

(2.5) in (1) is equivalent to , for all .

The following result will be used later.

holds for every subset of , then has a fixed point.

Motivated by [4, 5, 21], we give the following definition of mild solution of (1.1).

Definition 2.8.

Remark 2.9.

where .

We list the following basic assumptions of this paper.

where and .

(H4) For each , is measurable on and is bounded on . The map is continuous from to , here, .

where , .

## 3. Main Result

It follows from (H1), (H3), and (H4) that is well defined.

It will be shown that has a fixed point, and this fixed point is then a mild solution of (1.1).

Set , .

where .

Clearly, the operator has a fixed point is equivalent to has one. So, it turns out to prove that has a fixed point.

Now, we present and prove our main result.

Theorem 3.1.

Assume that (H1)–(H5) are satisfied, then there exists a mild solution of (1.1) on provided that .

Proof.

where .

This contradicts (2.17). Hence, for some positive number , .

This shows that is continuous.

It follows the continuity of in the uniform operator topology for that tends to 0, as . The continuity of ensures that tends to 0, as .

Clearly, the first term on the right-hand side of (3.20) tends to 0 as . The second term on the right-hand side of (3.20) tends to 0 as as a consequence of the continuity of in the uniform operator topology for .

Thus, is equicontinuous.

Now, let be an arbitrary subset of such that .

where . Therefore, .

Hence, . Applying now Lemma 2.7, we conclude that has a fixed point in . Let , then is a fixed point of the operator which is a mild solution of (1.1).

## 4. Application

where , , , , are continuous functions, and .

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

then we can see that in (2.6).

Then (4.1) can be reformulated as the abstract (1.1).

where , .

where .

then (4.1) has a mild solution by Theorem 3.1.

## Declarations

### Acknowledgments

The authors are grateful to the referees for their valuable suggestions. F. Li is supported by the NSF of Yunnan Province (2009ZC054M). J. Zhang is supported by Tianyuan Fund of Mathematics in China (11026100).

## Authors’ Affiliations

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