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

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
350
Issue
1
Identifiers
DOI: 10.1016/j.jmaa.2008.08.045
Keywords
  • Fixed Point
  • Approximate Fixed Point
  • Generalized Nonexpansive Mappings
  • James Constant
  • Nearly Uniformly Convex Space
  • Commuting Family
Disciplines
  • Mathematics

Abstract

Abstract 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.

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