ERGODIC RETRACTIONS FOR SEMIGROUPS IN STRICTLY CONVEX BANACH SPACES

• Published in: Taiwanese journal of mathematics Vol. 15; no. 4; pp. 1447 - 1456
• Main Authors: Kaczor, Wiesława, Reich, Simeon
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.08.2011
• Subjects: Banach space; Ergodic theory; Mathematical notation; Mathematical theorems; Semigroups; Topological compactness; Topological theorems; Topology
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/twjm/1500406356

We study the existence of ergodic retractions for semigroups of mappings in strictly convex Banach spaces. We prove, for instance, the following theorem. Let (X, ‖ · ‖) be a strictly convex Banach space and let Γ be a norming set forX. LetCbe a bounded and convex subset ofX, and supposeCis compact in the Γ-topology. If 𝒮 is a right amenable semigroup,φ= {Ts :s∈𝒮} is a semigroup onCwith a nonempty setF=F(φ) of common fixed points, and eachTs is (F-quasi-) nonexpansive, then there exists an (F-quasi-) nonexpansive retractionRfromContoFsuch thatRTs =TsR=Rfor eachs∈𝒮, and every Γ-closed, convex andφ-invariant subset ofCis alsoR-invariant. 2000Mathematics Subject Classification: 47H09, 47H10, 47H20. Key words and phrases: Γ-topology, Mean, Nonexpansive ergodic retraction, Nonexpansive mapping, Nonexpansive semigroup, Quasi-nonexpansive ergodic retraction, Quasi-nonexpansive mapping, Quasi-nonexpansive semigroup.