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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,) is called pseudo-almost periodic (pap) if f=g+h with Bohr-ap g and hPAP 0 , i.e., 1 t-a a t h(s)ds0 as t. For such h, H(t):= 0 t h(s)ds is pap iff there is bX with H-bPAP 0 ; special case: if f(t)0 as t, then F(t)= a t fds is asymptotic ap iff F(t) has a limit as t. For pap f=g+h the F is pap iff there is bX such that a t hds-bPAP 0 ; assumptions here: F bounded and X does not contain c 0 , or F([a,)) weakly relatively compact. With this a recent result of Ruess and Summers is generalized, answering a question of them: If f: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 X and F is bounded, and if further there is bX such that 0 t ϕds-b is a wap null-function, where f=apg+wap null-function ϕ.

MSC:
42A75Classical almost periodic functions, mean periodic functions
43A60Almost periodic functions on groups, etc.; almost automorphic functions
34C27Almost and pseudo-almost periodic solutions of ODE
34C28Complex behavior, chaotic systems (ODE)