- Research Article
- Open Access
© Claudio Cuevas et al. 2011
- Received: 23 September 2010
- Accepted: 8 December 2010
- Published: 14 December 2010
- Banach Space
- Periodic Function
- Fractional Order
- Bounded Function
- Mild Solution
-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  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 . 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 , Gorenflo and Mainardi  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 , Eidelman and Kochubei , Lakshmikantham et al. [27–30], Mophou and N'Guérékata , Ahmed and Nieto , and N'Guérékata . 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.
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
In order to give an operator theoretical approach for the study of the abstract system we recall the following definition.
Definition 2.1 (see ).
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 .
We recall the following definitions.
Definition 2.3 (see ).
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 ).
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 ).
To prove some of our results, we need the following lemma.
which proves the assertion.
which completes the proof.
Definition 3.1 (see ).
Now, we can establish our first existence result.
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.
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).
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.
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