Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 584874, 13 pages doi:10.1155/2011/584874 Research Article Weighted S-Asymptotically ω-Periodic Solutions of a Class of Fractional Differential Equations Claudio Cuevas, 1 Michelle Pierri, 2 and Alex Sepulveda 3 1 Departamento de Matem´ atica, Universidade Federal de Pernambuco, 50540-740 Recife, PE, Brazil 2 Departamento de F´ ısica e Matem´ atica da Faculdade de Filosofia, Ciˆ encias e Letras de Ribeir˜ ao Preto, Universidade de S˜ ao Paulo, 14040-901 Ribeir˜ ao Preto, SP, Brazil 3 Departamento de Matem´ atica y Estad´ ıstica, Universidad de La Frontera, Casilla 54-D, Temuco, Chile Correspondence should be addressed to Claudio Cuevas, cch@dmat.ufpe.br Received 23 September 2010; Accepted 8 December 2010 Academic Editor: J. J. Trujillo Copyright q 2011 Claudio Cuevas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence of weighted S-asymptotically ω-periodic mild solutions for a class of abstract fractional differential equations of the form u ′  ∂ −α1 Au  f t, u, 1 <α< 2, where A is a linear sectorial operator of negative type. 1. Introduction S-asymptotically ω-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 S-asymptotic ω-periodicity was introduced in the literature by Henr´ ıquez et al. 1, 2. Since then, it attracted the attention of many researchers see 1–10. In Pierri 10 a new S-asymptotically ω-periodic space was introduced. It is called the space of weighted S-asymptotically ω-periodic or Sv- asymptotically ω-periodic functions. In particular, the author has established conditions under which a Sv-asymptotically ω-periodic function is asymptotically ω-periodic and also discusses the existence of Sv-asymptotically ω-periodic solutions for an integral abstract Cauchy problem. The author has applied the results to partial integrodifferential equations. We study in this paper sufficient conditions for the existence and uniqueness of a weighted S-asymptotically ω-periodic mild solution to the following semi-linear integrodifferential equation of fractional order v ′ t  t 0 t − s α−2 Γα − 1 Avsds  f t, vt, t ≥ 0, 1.1