Asymptotic almost periodicity and motions of semigroups of operators
In this survey, we present on overview of some of our recent work [Math. Ann. 267, 145-158 (1986; Zbl 0584.54038), Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity, Math. Nachr., to appear, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Dissertationes Math., to appear] in resolving certain questions that arise in a qualitative study of solutions to the abstract Cauchy problem. Focusing on motions of strongly continuous semigroups of operators and their asymptotic behavior, we address (1) the characterization of asymptotically almost periodically motions and (2) the characterization of those asymptotically almost periodic functions for which this property carries over to their integral. Moreover, we sketch our solution of the problem raised by Nemytskii and Stepanov as to when the \(\omega\)-limit set of a motion of a dynamical system is a minimal set of almost periodic motions.
solutions to the abstract Cauchy problem
strongly continuous semigroups of operators
asymptotically almost periodically motions
scientific article; zbMATH DE number 3998601
scientific article