Fixed point properties for semigroups of nonlinear mappings and amenability

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 $S_1 = < a, b: ab = 1 >$.