Weighted S-Asymptotically ω-Periodic Solutions of a Class of Fractional Differential Equations

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 Henrı́quez 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 Svasymptotically 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