Nonexpansive mappings on Abelian Banach algebras and their fixed points
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutteaya Rd., Bangkok, 10400, Thailand
Fixed Point Theory and Applications 2012, 2012:150 doi:10.1186/1687-1812-2012-150Published: 14 September 2012
A Banach space X is said to have the fixed point property if for each nonexpansive mapping 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) for each such that for each , (iii) 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).
MSC: 46B20, 46J99.