×

zbMATH — the first resource for mathematics

Integration of vector-valued pseudo-almost periodic functions. (English) Zbl 0818.42003
A Banach space \(X\) valued bounded continuous function \(f\) on the interval \([a, \infty)\) is called pseudo-almost periodic (pap) if \(f= g+ h\) with Bohr-ap \(g\) and \(h\in PAP_ 0\), i.e., \({1\over t- a} \int^ t_ a \| h(s)\| ds\to 0\) as \(t\to \infty\). For such \(h\), \(H(t):= \int^ t_ 0 h(s) ds\) is pap iff there is \(b\in X\) with \(H- b\in PAP_ 0\); special case: if \(f(t)\to 0\) as \(t\to \infty\), then \(F(t)= \int^ t_ a f ds\) is asymptotic ap iff \(F(t)\) has a limit as \(t\to \infty\). For pap \(f= g+ h\) the \(F\) is pap iff there is \(b\in X\) such that \(\int^ t_ a h ds- b\in PAP_ 0\); assumptions here: \(F\) bounded and \(X\) does not contain \(c_ 0\), or \(F([a, \infty))\) weakly relatively compact. With this a recent result of Ruess and Summers is generalized, answering a question of them: If \(f:\mathbb{R}\to X\) is Eberlein weakly ap (wap), then the indefinite integral \(F\) is again wap iff either \(F(R)\) is weakly relatively compact, or \(c_ 0\not\subset X\) and \(F\) is bounded, and if further there is \(b\in X\) such that \(\int^ t_ 0 \varphi ds- b\) is a wap null-function, where \(f= \text{ap} g+ \text{wap}\) null-function \(\varphi\).
Reviewer: H.Günzler (Kiel)

MSC:
42A75 Classical almost periodic functions, mean periodic functions
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. · Zbl 0727.22001
[2] C. Corduneanu, Almost periodic functions, Chelsea, New York, 1989. · Zbl 0672.42008
[3] K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63 – 97. · Zbl 0104.05501
[4] M. I. Kadec, The integration of almost periodic functions with values in a Banach space, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 71 – 74 (Russian).
[5] W. M. Ruess and W. H. Summers, Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity, Dissertationes Math. (Rozprawy Mat.) 279 (1989), 38. · Zbl 0668.43005
[6] S. Zaidman, Almost-periodic functions in abstract spaces, Research Notes in Mathematics, vol. 126, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0648.42006
[7] C. Zhang, Pseudo almost periodic functions and their applications, thesis, University of Western Ontario, 1992.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.