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Zbl 1010.47032
Butnariu, D.; Reich, S.; Zaslavski, A.J.
Asymptotic behavior of relatively nonexpansive operators in Banach spaces. (English)
[J] J. Appl. Anal. 7, No.2, 151-174 (2001). ISSN 1425-6908; ISSN 1869-6082/e

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.
[Zhilin Yang (Qingdao, Shandong Province)]

MSC 2000:: *47H09 Mappings defined by "shrinking" properties
49M30 Methods of successive approximation, not based on necessary cond.
52A41 Convex functions and convex programs (convex geometry)

Keywords: Bregman distance; convex function; fixed point; generic property; iterative algorithm

Cited in: Zbl 1088.47054

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