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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)

##### 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
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