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Zbl 0818.42003
Zhang, Chuanyi
Integration of vector-valued pseudo-almost periodic functions. (English)
[J] Proc. Am. Math. Soc. 121, No.1, 167-174 (1994). ISSN 0002-9939; ISSN 1088-6826/e

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 $h\in PAP\sb 0$, i.e., ${1\over t- a} \int\sp t\sb a \Vert h(s)\Vert ds\to 0$ as $t\to \infty$. For such $h$, $H(t):= \int\sp t\sb 0 h(s) ds$ is pap iff there is $b\in X$ with $H- b\in PAP\sb 0$; special case: if $f(t)\to 0$ as $t\to \infty$, then $F(t)= \int\sp t\sb a f ds$ 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 $\int\sp t\sb a h ds- b\in PAP\sb 0$; assumptions here: $F$ bounded and $X$ does not contain $c\sb 0$, or $F([a, \infty))$ weakly relatively compact. With this a recent result of Ruess and Summers is generalized, answering a question of them: If $f:\bbfR\to X$ is Eberlein weakly ap (wap), then the indefinite integral $F$ is again wap iff either $F(R)$ is weakly relatively compact, or $c\sb 0\not\subset X$ and $F$ is bounded, and if further there is $b\in X$ such that $\int\sp t\sb 0 \varphi ds- b$ is a wap null-function, where $f= \text{ap} g+ \text{wap}$ null-function $\varphi$.
[H.Günzler (Kiel)]

MSC 2000:: *42A75 Periodic functions and generalizations
43A60 Almost periodic functions on groups, etc.
34C27 Almost periodic solutions of ODE
34C28 Other types of "recurrent" solutions of ODE

Keywords: pseudo-almost periodic functions; weakly almost periodic functions; integration; Eberlein weakly almost periodic functions

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