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Asymptotic behavior of relatively nonexpansive operators in Banach spaces. (English) Zbl 1010.47032
Let K be a closed convexed subset of a Banach space X, and let F be a nonempty closed subset of K. The authors consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. The aim of this paper is to prove that under quite mild conditions on F strong convergence of the sequences {T k x} k=1 generated by relatively nonexpansive mappings is the rule and that weak, but not strong convergence is the exception.

MSC:
47H09Mappings defined by “shrinking” properties
49M30Other numerical methods in calculus of variations
52A41Convex functions and convex programs (convex geometry)