On ultimately nonexpansive semigroups
Michael L. Edelstein, Mo Tak Kiang · Pacific Journal of Mathematics · 1982
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 / in G such that for all g in G, d(fg(u), fg(v)) ^ (1 + a)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.