Abstract
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.
Highlights
1 Introduction Almost periodicity and almost automorphy are attractive topics in the qualitative theory of differential equations due to their significance and applications in physics, mathematical biology, control theory and others. These concepts are generalized in various ways, say, pseudo almost periodicity (Zhang [1,2,3]), weighted pseudo almost periodicity (Diagana [4, 5]), pseudo almost automorphy (Liang, Xiao and Zhang [6, 7]), weighted pseudo almost automorphy (Blot et al [8]), etc. These concepts have been widely used in the investigation of ordinary differential equations, partial differential equations, functional differential equations and fractional differential equations
3.2 Related properties with respect to translation invariance Let us start with the construction of μ-paa (μ-pap) functions through convolution
As an application of the results obtained in the last section, we study the existence and uniqueness of μ-paa mild solutions of the following fractional differential equation: CDαt u(t) = Au(t) + CDαt –1f t, u(t), t ∈ R, 1 < α < 2, (4.1)
Summary
Almost periodicity and almost automorphy are attractive topics in the qualitative theory of differential equations due to their significance and applications in physics, mathematical biology, control theory and others. It known that two spaces of μ-paa (μ-pap) functions are the same if the measures of the spaces are equivalent.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
