Vol. 101, No. 1, 1982

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On ultimately nonexpansive semigroups

M. Edelstein and Mo Tak Kiang

Vol. 101 (1982), No. 1, 93–102

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.

Mathematical Subject Classification 2000
Primary: 47H10
Secondary: 54H25
Received: 12 December 1980
Published: 1 July 1982
M. Edelstein
Mo Tak Kiang