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].

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].

Reviewer: M.Sablik (Katowice)

##### 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 |