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

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, 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.

Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces

“…Specially, he proved [1,Theorem 5] that if K is a weakly compact convex subset of a uniformly convex in every direction (UCED) Banach space and if T : K → K is a mapping satisfying condition (C), then T has a fixed point. It was quickly noted by Dhompongsa et al [2] that one obtains the same conclusion if the domain K of T is a bounded closed and convex subset of a Banach space and every asymptotic center of a bounded sequence relative to K is nonempty and compact. Moreover, Dhompongsa and Kaewcharoen [3] extended Betiuk-Pilarska and Prus's result [4] on the weak fixed point property to continuous mappings satisfying condition (C) on an order uniformly noncreasy (OUNC) Banach lattice.…”

Fixed point theory for a class of generalized nonexpansive mappings

“…sequences, that is, it shares (i) with nonexpansive mappings (see [1,Lemma 6]), as well as (ii), because for some Banach spaces (see [1,Theorems 4,5]) mappings satisfying (C) leaving invariant weakly compact convex subsets have fixed points. (See also [4].) In this paper we define two kind of generalizations of condition (C).…”

Fixed points, selections and common fixed points for nonexpansive-type mappings

“…Dhompongsa et al [3] have recently proved the T invariance of the asymptotic center in K of an approximate fixed point sequence for T , when T is a singlevalued mapping satisfying condition (C). We now state a result which can be seen as an adaptation of this fact to the multivalued case.…”