*(English)*Zbl 0943.47040

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 $D(x,y)=f\left(x\right)-f\left(y\right)-{f}^{\text{'}\text{'}}\left(y\right)(x-y)\xb7$ For a selfmapping $T$ of a convex set $S\subset X$ denote by $\widehat{F}\left(T\right)$ the set of its asymptotic fixed points. $T$ is said to be strongly nonexpansive (with respect to a nonempty $\widehat{F}\left(T\right)$) if $D(p,T(x\left)\right)\le D(p,x)$ for all $p\in \widehat{F}\left(T\right)$ and $x\in S$ and if ${lim}_{n\to \infty}(D(p,{x}_{n})-D(p,T{x}_{n}))=0$ implies ${lim}_{n\to \infty}D(T{x}_{n},{x}_{n})=0$ for any $p\in \widehat{F}\left(T\right)$ and bounded sequence $\left({x}_{n}\right)$.

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

${lim}_{n\to \infty}{({T}_{m}{T}_{m-1}\cdots {T}_{1})}^{n}x$ 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 |