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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 $\left[a,\infty \right)$ is called pseudo-almost periodic (pap) if $f=g+h$ with Bohr-ap $g$ and $h\in PA{P}_{0}$, i.e., $\frac{1}{t-a}{\int }_{a}^{t}\parallel h\left(s\right)\parallel ds\to 0$ as $t\to \infty$. For such $h$, $H\left(t\right):={\int }_{0}^{t}h\left(s\right)ds$ is pap iff there is $b\in X$ with $H-b\in PA{P}_{0}$; special case: if $f\left(t\right)\to 0$ as $t\to \infty$, then $F\left(t\right)={\int }_{a}^{t}fds$ is asymptotic ap iff $F\left(t\right)$ has a limit as $t\to \infty$. For pap $f=g+h$ the $F$ is pap iff there is $b\in X$ such that ${\int }_{a}^{t}hds-b\in PA{P}_{0}$; assumptions here: $F$ bounded and $X$ does not contain ${c}_{0}$, or $F\left(\left[a,\infty \right)\right)$ weakly relatively compact. With this a recent result of Ruess and Summers is generalized, answering a question of them: If $f:ℝ\to X$ is Eberlein weakly ap (wap), then the indefinite integral $F$ is again wap iff either $F\left(R\right)$ is weakly relatively compact, or ${c}_{0}\text{⊄}X$ and $F$ is bounded, and if further there is $b\in X$ such that ${\int }_{0}^{t}\varphi ds-b$ is a wap null-function, where $f=\text{ap}g+\text{wap}$ null-function $\varphi$.
##### MSC:
 42A75 Classical almost periodic functions, mean periodic functions 43A60 Almost periodic functions on groups, etc.; almost automorphic functions 34C27 Almost and pseudo-almost periodic solutions of ODE 34C28 Complex behavior, chaotic systems (ODE)