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It is well known that many important properties of μ-pseudo almost automorphic (periodic) functions are based on the translation invariance. In this paper we give a new result of the translation invariance with conditions weaker than the known one. Then some more properties are studied, including the uniqueness of the decomposition and the convolution of the space. Moreover, we present a result on the equality of two spaces of this kind of functions, which extends the well-known results to the case when the measures of the spaces are not equivalent. As an application, we give an existence and uniqueness theorem of μ-pseudo almost automorphic mild solution for a semilinear fractional differential equation. [ABSTRACT FROM AUTHOR]

Our aim in the article is to study the existence of μ-pseudo almost automorph mild solutions to the following fractional integro-differential equation: Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t,u(t)),t∈R, where for α>0, the fractional derivative Dα is understood in the sense of Weyl, and A is a closed linear operator defined on Banach space X, a∈Lloc1(R+) is a scalar-valued kernel. The novelty of this work is a study of this equation with a μ-Sp-pseudo almost automorph nonlinear term satisfying the condition of “uniform continuity” instead of some “Lipschitz” type conditions supposed in the literature. We utilize Schauder’s fixed point theorem. An example is provided to explain our abstract results. [ABSTRACT FROM AUTHOR]