Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

• Main Authors: Dhompongsa S., Inthakon W., Kaewkhao A.
• Format: Article
• Language: English
• Published: 2014
• Online Access: http://www.scopus.com/inward/record.url?eid=2-s2.0-54149110520&partnerID=40&md5=5636e4f6a1d506045fc5400255509c93; http://cmuir.cmu.ac.th/handle/6653943832/5938
• Institution: Chiang Mai University

A new condition for mappings, called condition (C), which is more general than nonexpansiveness, was recently introduced by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. Following the idea of Kirk and Massa Theorem in [W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 364-375], we prove a fixed point theorem for mappings with condition (C) on a Banach space such that its asymptotic center in a bounded closed and convex subset of each bounded sequence is nonempty and compact. This covers a result obtained by Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095]. We also present fixed point theorems for this class of mappings defined on weakly compact convex subsets of Banach spaces satisfying property (D). Consequently, we extend the results in [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095] to many other Banach spaces. © 2008 Elsevier Inc. All rights reserved.