Abstract: The bounded primitive of a periodic function is periodic, and the bounded primitive of an asymptotically periodic function is not necessarily asymptotically periodic. In this paper, we introduce the concept of slowly periodic functions and prove that the bounded primitive function of an asymptotically periodic function is slowly periodic. Interestingly, slowly periodic functions are just a special class of $S$-asymptotically periodic functions, which were introduced 15 years ago and extensively studied in recent years. On this basis, a Tauberian theorem for asymptotically periodic functions and two related Tauberian theorems are established. Moreover, we apply our Tauberian theorems to the nonhomogeneous abstract Cauchy problem, and obtain the spectral condition under which the solution of Cauchy problem is $\S$-asymptotically periodic. In our Tauberian theorem for asymptotically periodic functions and its application to abstract Cauchy problem, we completely remove the ergodic assumption in [J. Differential Equations 122 (1995), 282-301] although the conclusions are slightly weaker than asymptotical periodicity. Finally, we construct a concrete Cauchy problem as an example. It is worth mentioning that the inhomogeneous term of this Cauchy problem is asymptotically periodic and its solution is $\S$-asymptotically periodic rather than asymptotically periodic. This demonstrates that $\S$-asymptotically periodic functions are the ``natural class'' for solutions to some differential equations.

Key words: asymptotically periodic, slowing periodic, S-asymptotically periodic, abstract Cauchy problem, Tauberian theorem, Beurling spectrum