# 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

**DOI: **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.

## 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)If is continuous on and , then and is continuous in and(2.5)
where is a constant.

- (2)There exist a continuous function and a locally bounded function in such that(2.6)
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

## References

- Agarwal RP, de Andrade B, Cuevas C:
**On type of periodicity and ergodicity to a class of fractional order differential equations.***Advances in Difference Equations*2010,**2010:**-25.Google Scholar - Ahmed HM:
**Boundary controllability of nonlinear fractional integrodifferential systems.***Advances in Difference Equations*2010,**2010:**-9.Google Scholar - Alsaedi A, Ahmad B:
**Existence of solutions for nonlinear fractional integro-differential equations with three-point nonlocal fractional boundary conditions.***Advances in Difference Equations*2010,**2010:**-10.Google Scholar - El-Borai MM:
**Some probability densities and fundamental solutions of fractional evolution equations.***Chaos, Solitons and Fractals*2002,**14**(3):433-440. 10.1016/S0960-0779(01)00208-9MathSciNetView ArticleMATHGoogle Scholar - El-Borai MM:
**On some stochastic fractional integro-differential equations.***Advances in Dynamical Systems and Applications*2006,**1**(1):49-57.MathSciNetMATHGoogle Scholar - Mophou GM, N'Guérékata GM:
**Existence of the mild solution for some fractional differential equations with nonlocal conditions.***Semigroup Forum*2009,**79**(2):315-322. 10.1007/s00233-008-9117-xMathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J, van Casteren J:
**A note on semilinear abstract functional differential and integrodifferential equations with infinite delay.***Applied Mathematics Letters*2004,**17**(4):473-477. 10.1016/S0893-9659(04)90092-4MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao T-J:
**Semilinear integrodifferential equations with nonlocal initial conditions.***Computers & Mathematics with Applications*2004,**47**(6-7):863-875. 10.1016/S0898-1221(04)90071-5MathSciNetView ArticleMATHGoogle Scholar - Liang J, Liu JH, Xiao T-J:
**Nonlocal problems for integrodifferential equations.***Dynamics of Continuous, Discrete & Impulsive Systems. Series A*2008,**15**(6):815-824.MathSciNetMATHGoogle Scholar - Xiao T-J, Liang J:
**Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(12):e1442-e1447. 10.1016/j.na.2009.01.204MathSciNetView ArticleMATHGoogle Scholar - Hale JK, Kato J:
**Phase space for retarded equations with infinite delay.***Funkcialaj Ekvacioj*1978,**21**(1):11-41.MathSciNetMATHGoogle Scholar - Hernández E, Henríquez HR:
**Existence results for partial neutral functional-differential equations with unbounded delay.***Journal of Mathematical Analysis and Applications*1998,**221**(2):452-475. 10.1006/jmaa.1997.5875MathSciNetView ArticleMATHGoogle Scholar - Liang J, Xiao TJ:
**Functional-differential equations with infinite delay in Banach spaces.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):497-508. 10.1155/S0161171291000686MathSciNetView ArticleMATHGoogle Scholar - Mophou GM, N'Guérékata GM:
**Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay.***Applied Mathematics and Computation*2010,**216**(1):61-69. 10.1016/j.amc.2009.12.062MathSciNetView ArticleMATHGoogle Scholar - Mophou GM, N'Guérékata GM:
**A note on a semilinear fractional differential equation of neutral type with infinite delay.***Advances in Difference Equations*2010,**2010:**-8.Google Scholar - Hernández E, Henríquez HR:
**Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay.***Journal of Mathematical Analysis and Applications*1998,**221**(2):499-522. 10.1006/jmaa.1997.5899MathSciNetView ArticleMATHGoogle Scholar - Banaś J, Goebel K:
*Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics*.*Volume 60*. Marcel Dekker, New York, NY, USA; 1980:vi+97.Google Scholar - Heinz H-P:
**On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions.***Nonlinear Analysis: Theory, Methods & Applications*1983,**7**(12):1351-1371. 10.1016/0362-546X(83)90006-8MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Meehan M, O'Regan D:
*Fixed Point Theory and Applications, Cambridge Tracts in Mathematics*.*Volume 141*. Cambridge University Press, Cambridge, UK; 2001:x+170.View ArticleMATHGoogle Scholar - Szufla S:
**On the application of measure of noncompactness to existence theorems.***Rendiconti del Seminario Matematico della Università di Padova*1986,**75:**1-14.MathSciNetMATHGoogle Scholar - Zhou Y, Jiao F:
**Nonlocal Cauchy problem for fractional evolution equations.***Nonlinear Analysis: Real World Applications*2010,**11**(5):4465-4475. 10.1016/j.nonrwa.2010.05.029MathSciNetView ArticleMATHGoogle Scholar - Mainardi F, Paradisi P, Gorenflo R:
**Probability distributions generated by fractional diffusion equations.**In*Econophysics: An Emerging Science*. Edited by: Kertesz J, Kondor I. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000.Google Scholar - Bothe D:
**Multivalued perturbations of**m**-accretive differential inclusions.***Israel Journal of Mathematics*1998,**108:**109-138. 10.1007/BF02783044MathSciNetView ArticleMATHGoogle Scholar

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