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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 $$\{ T^k x\}_{k=1}^\infty$$ generated by relatively nonexpansive mappings is the rule and that weak, but not strong convergence is the exception.

##### MSC:
 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 49M30 Other numerical methods in calculus of variations (MSC2010) 52A41 Convex functions and convex programs in convex geometry
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