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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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