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Ruess, W. M.; Summers, W. H.

Asymptotic almost periodicity and motions of semigroups of operators. (English) Zbl 0616.47047

Linear Algebra Appl. 84, 335-351 (1986).
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.

Cited in 17 Documents

MSC:

47H20 Semigroups of nonlinear operators
47D03 Groups and semigroups of linear operators
58D07 Groups and semigroups of nonlinear operators
46E40 Spaces of vector- and operator-valued functions
46A50 Compactness in topological linear spaces; angelic spaces, etc.

Keywords:

solutions to the abstract Cauchy problem; strongly continuous semigroups of operators; asymptotic behavior; asymptotically almost periodically motions

Citations:

Zbl 0584.54038
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Full Text: DOI

References:

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