  In a series 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 particular, we introduced the
notion of semihypergroup actions on a general topological space and discussed
different continuity, equivalence and natural fixed point properties of the
same in [6]. Now in this article, we consider different kinds of
representations of a semihypergroup on compact convex subsets of a locally
convex space and explore equivalence relations between certain fixed-point
properties of such representations and amenability of the space of almost
periodic functions. Finally, we investigate how far these equivalence relations
can be strengthened when in particular, we consider representations on the dual
of a Banach space.
