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, 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.
Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and T : E → K(X) a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p ∈ E such that αp ⊕ (1 − α)T x ⊂ I E (x) ∀x ∈ E, ∀α ∈ [0, 1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.
The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan-von Neumann constant C NJ (X) of a Banach space X. Using this fact, we prove that if C NJ (X) is less than an appropriate positive number, then every multivalued nonexpansive mapping T : E → KC(E) has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.
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