Fixed point theorem for nonexpansive semigroups on Banach space. (English) Zbl 0818.47055

Takahashi, Wataru; Jeong, Doo Hoan

Proc. Am. Math. Soc. 122, No. 4, 1175-1179 (1994).

Summary: Let \(C\) be a nonempty closed convex subset of a uniformly convex Banach space, and let \(S\) be a semitopological semigroup such that \(\text{RUC}(S)\) has a left invariant submean. Then we prove a fixed point theorem for a continuous representation of \(S\) as nonexpansive mappings on \(C\).

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H20	Semigroups of nonlinear operators

Keywords:

nonempty closed convex subset of a uniformly convex Banach space; semitopological semigroup; left invariant submean; fixed point theorem; continuous representation; nonexpansive mappings