On the asymptotics of (c)-mapping iterations
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Journal of Fixed Point Theory and Applications
On the asymptotics of (c)-mapping iterations Abdelkader Dehici and Najeh Redjel Abstract. In this paper, we give an investigation of the asymptotics for the iterations associated with (c)-mappings acting on unbounded closed convex subsets of Banach spaces. In particular, the almost fixed point property (in short; AFPP) and conditions ensuring an ergodic type theorem for this class of mappings are established. Mathematics Subject Classification. 47H10, 54H25. Keywords. Banach space, (c)-mapping, unbounded closed convex subset, linearly bounded subset, weakly compact, iteration, almost fixed point sequence, asymptotic radius, asymptotic center, AFPP property, strictly convex Banach space, uniformly convex Banach space, fixed point.
1. Introduction and preliminaries Since the work of Browder, G¨ ohde and Kirk (see [7,13,17]), the link between the existence of fixed points for nonexpansive mappings and the geometry of Banach spaces was discovered. This theory has been flourished due to the contributions of many authors who developed the subject by studying various geometrical notions and tools in Banach spaces. As a consequence of this upgrowth, several unsolved problems for a long time have been answered. For a good reading on this area, we can quote [1,3,6,11,12,16–19,28,29,34,36] and the references therein. The goal of these investigations is to apply these theoretic methods for the setting of differential equations, partial differential equations and different branches of analysis. For example, it was proved that holomorphic mappings are nonexpansive with respect to certain pseudometrics (see Chapter 2 in [12] and the book by Reich and Shoikhet [32]). One of the most important subjects which appears in these works is the asymptotics of iterations associated with a given mapping even if it has a fixed point since for the case of the nonexpansive mappings, Picard sequences do not necessarily converge or weakly converge to a fixed point. The study of this major question is not an easy task and many problems related to it are still open. In this direction, we quote the contributions of Reich and his collaborators (see [21,25–27,33]). We can also find in [2], a fine analysis on 0123456789().: V,-vol
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A. Dehici and N. Redjel
the asymptotics of iterations associated with firmly nonexpansive mappings in W-hyperbolic spaces and CAT(0) spaces. Now, we ask the following: What happens to the other contractions? In [35], Smyth asked whether the FPP (fixed point property) for nonexpansive mappings and the FPP for (c)-mappings are closely connected when these mappings are defined on bounded closed and convex subsets of an arbitrary Banach space and he gave some situations where these two properties hold simultaneously. But the answer to this question in the case of unbounded sets is still unknown. Throughout this paper, (X, .) is a real Banach space, X denotes its dual and C is a nonempty closed subset of X. Definition 1.1. Let T : C −→ C be a self-mapping. (i) T is said t
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