## 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