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A weak convergence theorem for the alternating method with Bregman distances. (English) Zbl 0943.47040
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).

Let $X$ be a reflexive Banach space, and let$f:X\to ℝ$ be a convex continuous functional which is Gǎteaux differentiable. The Bregman distance corresponding to $f$ is defined by $D\left(x,y\right)=f\left(x\right)-f\left(y\right)-{f}^{\text{'}\text{'}}\left(y\right)\left(x-y\right)·$ For a selfmapping $T$ of a convex set $S\subset X$ denote by $\stackrel{^}{F}\left(T\right)$ the set of its asymptotic fixed points. $T$ is said to be strongly nonexpansive (with respect to a nonempty $\stackrel{^}{F}\left(T\right)$) if $D\left(p,T\left(x\right)\right)\le D\left(p,x\right)$ for all $p\in \stackrel{^}{F}\left(T\right)$ and $x\in S$ and if ${lim}_{n\to \infty }\left(D\left(p,{x}_{n}\right)-D\left(p,T{x}_{n}\right)\right)=0$ implies ${lim}_{n\to \infty }D\left(T{x}_{n},{x}_{n}\right)=0$ for any $p\in \stackrel{^}{F}\left(T\right)$ and bounded sequence $\left({x}_{n}\right)$.

The main result states the following. If ${T}_{j},j\in \left\{1,\cdots ,m\right\}$ are strongly nonexpansive self-mappings of a convex set $S\subset X$, the intersection $F$ of $\stackrel{^}{F}\left({T}_{j}\right),j\in \left\{1,\cdots ,m\right\}$ as well as $\stackrel{^}{F}\left({T}_{m}{T}_{m-1}\cdots {T}_{1}\right)$ are nonempty and ${f}^{\text{'}\text{'}}$ is weakly sequentially continuous then the weak

${lim}_{n\to \infty }{\left({T}_{m}{T}_{m-1}\cdots {T}_{1}\right)}^{n}x$ exists for each $x\in S$ and belongs to $F$.

Applications to convex sets intersection problem and to finding a common zero of finitely many monotone operators are given.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H07 Monotone and positive operators on ordered topological linear spaces 46B10 Duality and reflexivity in normed spaces