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Zbl 0943.47040
Reich, Simeon
A weak convergence theorem for the alternating method with Bregman distances.
(English)
[A] Kartsatos, Athanassios G. (ed.), Theory and applications of nonlinear operators of accretive and monotone type. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 178, 313-318 (1996). ISBN 0-8247-9721-3/pbk

Let $X$ be a reflexive Banach space, and let$f:X\to\bbfR$ be a convex continuous functional which is Gǎteaux differentiable. The Bregman distance corresponding to $f$ is defined by $D(x,y) =f(x) - f(y) - f''(y)(x-y).$ For a selfmapping $T$ of a convex set $S\subset X$ denote by $\hat{F}(T)$ the set of its asymptotic fixed points. $T$ is said to be strongly nonexpansive (with respect to a nonempty $\hat{F}(T)$) if $D(p,T(x))\leq D(p,x)$ for all $p\in \hat{F}(T)$ and $x\in S$ and if $\lim_{n\to \infty} (D(p,x_n)-D(p,Tx_n)) =0$ implies $\lim_{n\to \infty} D(Tx_n,x_n)=0$ for any $p\in \hat{F}(T)$ and bounded sequence $(x_n)$.\par The main result states the following. If $T_j, j\in \{1,\dots ,m\}$ are strongly nonexpansive self-mappings of a convex set $S\subset X$, the intersection $F$ of $\hat{F}(T_j), j\in \{1,\dots ,m\}$ as well as $\hat{F}(T_mT_{m-1}\dots T_1)$ are nonempty and $f''$ is weakly sequentially continuous then the weak\par $\lim_{n\to \infty}(T_mT_{m-1}\dots T_1)^nx$ exists for each $x\in S$ and belongs to $F$.\par Applications to convex sets intersection problem and to finding a common zero of finitely many monotone operators are given.
[M.Sablik (Katowice)]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H07 Positive operators on ordered topological linear spaces
46B10 Duality and reflexivity in normed spaces

Keywords: reflexive space; Bregman distance; Gǎteaux differentiability; iterative procedure; asymptotic fixed points; strongly nonexpansive self-mappings; convex sets intersection problem; common zero of finitely many monotone operators

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