Fixed point properties of semigroups of non-expansive mappings
By Anthony To-Ming Lau, Yong Zhang. 2008
90 citations recorded by the source corpus.
Abstract In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d006a242d2b15063c229c52b71ad143a1c78894e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Reiter nets for semidirect products of amenable groups and semigroups (2009, background): “Explicit constructions of Reiter nets for bicyclic semigroups can be found in [13].”
- Edelstein's method and fixed point theorems for some generalized nonexpansive mappings (2009, result): “Inspired by the results in [11,15,16], we finish the paper with the following question:”
- Fixed point properties for semigroups of nonlinear mappings and amenability (2012, background): “It is also well-known that if AP(S), the space of continuous almost periodic functions on S, has a left invariant mean, C is compact, convex and S is a nonexpansive representation of S on C, then F(S) = ∅ (see [24,27,32]).”