  In this paper we study fixed point properties for semitopological semigroup
of nonexpansive mappings on a bounded closed convex subset of a Banach space.
We also study a Schauder fixed point property for a semitopological semigroup
of continuous mappings on a compact convex subset of a separated locally convex
space. Such semigroups properly include the class of extremely left amenable
semitopological semigroups, the free commutative semigroup on one generator and
the bicyclic semigroup $S_1 = < a, b: ab = 1 >$.
