The τ-fixed point property for left reversible semigroups

Francisco E Castillo-Santos,Maria A Japón

doi:10.1186/s13663-015-0357-7

Abstract

In this article we use the generalized Gossez-Lami Dozo property and the Opial condition to study the fixed point property for left reversible semigroups in separable Banach spaces. As a consequence, some previous results will be deduced and new examples of Banach spaces satisfying the fixed point property for left reversible semigroups are shown. We will also extend some previous theorems when we consider the semigroup formed by a unique nonexpansive mapping and its iterates.

Highlights

1 Introduction A semigroup S is said to be a semitopological semigroup if S is equipped with a Hausdorff topology such that for each a ∈ S, the two mappings from S into S defined by s → as and s → sa are continuous
Let C be a subset of a Banach space X and let S be a semitopological semigroup
In this paper we develop new arguments to deduce whether a dual Banach space satisfies the weak∗-FPP for left reversible semigroups

Summary

Introduction

A semigroup S is said to be a semitopological semigroup if S is equipped with a Hausdorff topology such that for each a ∈ S, the two mappings from S into S defined by s → as and s → sa are continuous. Particular examples of dual Banach spaces are known to satisfy the weak∗-FPP for left reversible semigroups. In Lim [ ] proved that the sequence space satisfies the weak∗-FPP for left reversible semigroups. We will consider τ as any translation invariant topology on a separable Banach space X and we give sufficient conditions to assure the τ -FPP for left reversible semigroups. We will extend some known results for nonexpansive mappings to the setting of the fixed point property for left reversible semigroups. Many examples of Banach spaces are known to satisfy the GGLD condition or the Opial property with respect to some classical topologies. Let X be a separable Banach space, τ be a translation invariant topology on X such that τ -compact sets are τ -sequentially compact. In [ ] (Lemma ) it is proved that the norm and the τ -null type functions are τ -slsc whenever they are constant on spheres

First examples and applications for the weak-star topology

A Musielak-Orlicz sequence space generated by is defined by

Modular function spaces and the ρ-FPP for left reversible semigroups