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Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations. (English) Zbl 1186.43008
A bounded and continuous function $f$ on the real line with values in a Banach space is a pseudo almost periodic function if it can be represented as $f=g+h,$ where $g$ is almost periodic and $h$ satisfies ${lim}_{|r|\to \infty }{\int }_{-r}^{r}\parallel h\left(t\right)\parallel \phantom{\rule{0.166667em}{0ex}}dt=0$. In the same way, the definition of pseudo almost automorphic functions is given. In this paper the authors study sufficient conditions for several composition operators to map a pseudo almost periodic function to a pseudo almost periodic function, or a pseudo almost automorphic function to a pseudo almost automorphic function. The results are then applied to the heat equation.
##### MSC:
 43A60 Almost periodic functions on groups, etc.; almost automorphic functions 34C27 Almost and pseudo-almost periodic solutions of ODE