  In a couple of previous papers, we initiated a systematic study of
semihypergroups and had a thorough discussion on certain analytic and algebraic
aspects associated to this class of objects. In this article, we introduce and
examine (separately) continuous actions on the category of semihypergroups. In
particular, we discuss the continuity properties of such actions and explore
the equivalence relations between different fixed-point properties of certain
actions and the existence of left-invariant mean(s) on the space of almost
periodic functions on a semihypergroup.
