Nonexpansive mappings on Abelian Banach algebras and their fixed points

• Main Author: Fupinwong W.
• Format: Journal
• Published: 2017
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84902528913&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/42911
• Institution: Chiang Mai University

A Banach space X is said to have the fixed point property if for each nonexpansive mapping T : E → E on a bounded closed convex subset E of X has a fixed point. We show that each infinite dimensional Abelian complex Banach algebra X satisfying: (i) property (A) defined in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010), (ii) ||x|| ≤ ||y|| for each x, y ∈ X such that |τ(x)| ≤ |τ(y)| for each τ ∈ Ω(X), (iii) inf{r(x) : x ∈ X, ||x|| = 1} > 0 does not have the fixed point property. This result is a generalization of Theorem 4.3 in (Fupinwong and Dhompongsa in Fixed Point Theory Appl. 2010:Article ID 34959, 2010). © 2012 Fupinwong; licensee Springer.