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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$$f:X\to\mathbb{R}$$ 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 $$(x_n)$$.
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
$$\lim_{n\to \infty}(T_mT_{m-1}\dots T_1)^nx$$ 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.
For the entire collection see [Zbl 0840.00034].

##### MSC:
 47H10 Fixed-point theorems 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 46B10 Duality and reflexivity in normed linear and Banach spaces