A semigroup G of continuous
selfmappings of a metric space (X,d) is called ultimately nonexpansive if
for every u, v in X and α > 0, there is an f in G such that for all g in G,
d(fg(u),fg(v)) ≦ (1 + α)d(u,v). It is shown that if G is an ultimately nonexpansive
commutative semigroup of selfmappings, then G has a fixed point when any one of
the following conditions is satisfied: (1) X is a reflexive Banach space and
each orbit under G is precompact; (2) X is a finite dimensional Banach
space and there is a point in X with a bounded orbit; (3) X is a reflexive,
locally uniformly convex Banach space having a point with a precompact
orbit.
