  The main purpose of this paper is to introduce the notion of an
asymptotically almost periodic ultradistribution and asymptotically almost
automorphic ultradistribution with values in a Banach space, as well as to
further analyze the classes of asymptotically almost periodic and
asymptotically almost automorphic distributions with values in a Banach space.
We provide some applications of the introduced concepts in the analysis of
systems of ordinary differential equations.
