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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)

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
Full Text: DOI
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