- Research Article
- Open Access

- Claudio Cuevas
^{1}Email author, - Michelle Pierri
^{2}and - Alex Sepulveda
^{3}

**2011**:584874

https://doi.org/10.1155/2011/584874

© Claudio Cuevas et al. 2011

**Received:**23 September 2010**Accepted:**8 December 2010**Published:**14 December 2010

## Abstract

## Keywords

- Banach Space
- Periodic Function
- Fractional Order
- Bounded Function
- Mild Solution

## 1. Introduction

-asymptotically -periodic functions have applications to several problems, for example in the theory of functional differential equations, fractional differential equations, integral equations and partial differential equations. The concept of -asymptotic -periodicity was introduced in the literature by Henríquez et al. [1, 2]. Since then, it attracted the attention of many researchers (see [1–10]). In Pierri [10] a new -asymptotically -periodic space was introduced. It is called the space of weighted -asymptotically -periodic (or -asymptotically -periodic) functions. In particular, the author has established conditions under which a -asymptotically -periodic function is asymptotically -periodic and also discusses the existence of -asymptotically -periodic solutions for an integral abstract Cauchy problem. The author has applied the results to partial integrodifferential equations.

where , is a linear densely defined operator of sectorial type on a complex Banach space and is an appropriate function. Note that the convolution integral in (1.1) is known as the Riemann-Liouville fractional integral [11]. We remark that there is much interest in developing theoretical analysis and numerical methods for fractional integrodifferential equations because they have recently proved to be valuable in various fields of sciences and engineering. For details, including some applications and recent results, see the monographs of Ahn and MacVinish [12], Gorenflo and Mainardi [13] and Trujillo et al. [14–16] and the papers of Agarwal et al. [17–23], Cuesta [11, 24], Cuevas et al. [5, 6], dos Santos and Cuevas [25], Eidelman and Kochubei [26], Lakshmikantham et al. [27–30], Mophou and N'Guérékata [31], Ahmed and Nieto [32], and N'Guérékata [33]. In particular equations of type (1.1) are attracting increasing interest (cf. [5, 11, 24, 34]).

The existence of weighted -asymptotically -periodic (mild) solutions for integrodifferential equation of fractional order of type (1.1) remains an untreated topic in the literature. Anticipating a wide interest in the subject, this paper contributes in filling this important gap. In particular, to illustrate our main results, we examine sufficient conditions for the existence and uniqueness of a weighted -asymptotically -periodic mild solution to a fractional oscillation equation.

## 2. Preliminaries and Basic Results

In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper. Let and be Banach spaces. The notation stands for the space of bounded linear operators from into endowed with the uniform operator topology denoted , and we abbreviate to and whenever . In this paper denotes the Banach space consisting of all continuous and bounded functions from into with the norm of the uniform convergence. For a closed linear operator we denote by the resolvent set and by the spectrum of (that is, the complement of in the complex plane). Set the resolvent of for .

### 2.1. Sectorial Linear Operators and the Solution Operator for Fractional Equations

A closed and linear operator
is said sectorial of type *μ* if there are
and
such that the spectrum of
is contained in the sector
and
, for all
.

In order to give an operator theoretical approach for the study of the abstract system we recall the following definition.

Definition 2.1 (see [17]).

Let be a closed linear operator with domain in a Banach space . One calls the generator of a solution operator for (1.1)-(1.2) if there are and a strongly continuous function such that and , for all . In this case, is called the solution operator generated by . By [35, Proposition 2.6], . We observe that the power function is uniquely defined as , with .

*μ*with , then is the generator of a solution operator given by , , where is a suitable path lying outside the sector (cf. [11]). Recently, Cuesta [11, Theorem 1] proved that if is a sectorial operator of type for some and , then there exists such that

Remark 2.2.

In the remainder of this paper, we always assume that is a a sectorial of type and , are the constants introduced above.

### 2.2. Weighted -Asymptotically -Periodic Functions

We recall the following definitions.

Definition 2.3 (see [1]).

A function is called -asymptotically -periodic if there exists such that . In this case, we say that is an asymptotic period of .

Throughout this paper, represents the space formed for all the -valued -asymptotically -periodic functions endowed with the uniform convergence norm denoted . It is clear that is a Banach space (see [1, Proposition 3.5]).

Definition 2.4 (see [10]).

Let . A function is called weighted -asymptotically -periodic (or -asymptotically -periodic) if .

Proposition 2.5.

Proof.

Let be a Cauchy sequence in . From the definition of , there exists such that in . Next, we prove that in .

which implies that for and as .

which implies that . This completes the proof.

Definition 2.6.

A function is called uniformly -asymptotically -periodic on bounded sets if for every bounded subset , the set is bounded and , uniformly for . If we say that is uniformly -asymptotically -periodic on bounded sets (see [1]).

To prove some of our results, we need the following lemma.

Lemma 2.7.

If , then the function belongs to .

Proof.

which proves the assertion.

Lemma 2.8.

Proof.

which completes the proof.

## 3. Existence of Weighted -Asymptotically -Periodic Solutions

In this section we discuss the existence of weighted -asymptotically -periodic solutions for the abstract system (1.1)-(1.2). To begin, we recall the definition of mild solution for (1.1)-(1.2).

Definition 3.1 (see [5]).

Now, we can establish our first existence result.

Theorem 3.2.

If , then there exits a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Suppose, there is a function such that and , for every and all . If is such that as , then is weighted -asymptotically -periodic.

Proof.

which implies that as , and hence . Moreover, from the above estimate it is easy to infer that , for all , is a contraction and there exists a unique -asymptotically -periodic mild solution of (1.1)-(1.2).

which shows that . This completes the proof.

Example 3.3.

whence is -asymptotically -periodic on bounded sets. By Theorem 3.2 we conclude that if , then there is a unique -asymptotically -periodic mild solution of (1.1)-(1.2). Moreover .

Theorem 3.4.

Proof.

for all , which shows that is a contraction on and hence there is a unique -asymptotically -periodic mild solution. The proof is complete.

where and . In what follows we consider the space and let be the operator given by with domain , . It is well known that is sectorial of type negative.

Proposition 3.5.

Let satisfying conditions of Lemma 2.8 and let . If is small enough, then the problems (3.20)–(3.22) has a unique -asymptotically -periodic mild solution.

Proof.

If we choose small enough, we have that condition (3.11) is fulfilled. By Theorem 3.4, the problems (3.20)–(3.22) has a unique -asymptotically -periodic (mild) solution. This finishes the proof.

## Declarations

### Acknowledgments

C. Cuevas thanks the Department of Mathematics of Universidad de La Frontera, where this project was started. The authors are grateful to the referees for their valuable comments and suggestions. C. Cuevas is partially supported by CNPQ/Brazil under Grant 300365/2008-0.

## Authors’ Affiliations

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