Almost periodic type functions and densities

Abstract

<p style='text-indent:20px;'>In this paper, we introduce and analyze the notions of <inline-formula><tex-math id="M1">\begin{document}$ \odot_{g} $\end{document}</tex-math></inline-formula>-almost periodicity and Stepanov <inline-formula><tex-math id="M2">\begin{document}$ \odot_{g} $\end{document}</tex-math></inline-formula>-almost periodicity for functions with values in complex Banach spaces. In order to do that, we use the recently introduced notions of lower and upper (Banach) <inline-formula><tex-math id="M3">\begin{document}$ g $\end{document}</tex-math></inline-formula>-densities. We also analyze uniformly recurrent functions, generalized almost automorphic functions and apply our results in the qualitative analysis of solutions of inhomogeneous abstract integro-differential inclusions. We present plenty of illustrative examples, results of independent interest, questions and unsolved problems.</p>