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

##### MSC:
 47H09 Mappings defined by “shrinking” properties 49M30 Other numerical methods in calculus of variations 52A41 Convex functions and convex programs (convex geometry)