SEMI-ASYMPTOTIC NON-EXPANSIVE ACTIONS OF SEMI-TOPOLOGICAL SEMIGROUPS

Abstract

Abstract. In this paper we extend Takahashi’s fixed point theorem ondiscrete semigroups to general semi-topological semigroups. Next we de-fine the semi-asymptotic non-expansive action of semi-topological semi-groups to give a partial affirmative answer to an open problem raised byA.T-M. Lau. 1. IntroductionA (not necessarily linear) self-mapping T : E → E on a Banach space Eis called non-expansive if kT(x) − T(y)k ≤ kx − yk for all x,y ∈ E. In 1963R. DeMarr proved the following common fixed point theorem for commutingfamilies of non-expansive self-mappings [6].Theorem 1.1. For any non-empty compact convex subset K of a Banach spaceE each commuting family of non-expansive self mappings on K has a commonfixed point in K.This generalizes the celebrated Markov-Kakutani fixed point theorem (forthe case of Banach spaces) on commuting families of continuous linear trans-formations on Hausdorff topological vector spaces leaving certain nonemptycompact convex subset invariant. DeMarr’s theorem has been generalized inseveral directions by Belluce and Kirk [2,3], Takahashi [16], Mitchell [14], Lauand Holmes [7,8].DeMarr theorem suggests that the action of certain commutative semigrouphas a fixed point. It is then natural to seek the same type offixed point propertyfor the actions of more general semigroups.Let S be a semi-topological semigroup, that is a semigroup with a Hausdorfftopology with separately continuous multiplication. We say that S is rightreversible if it has finite intersection property for closed left ideals. An actionof S on a topological space E is a mapping (s,x) → s(x) from S × E into