*(English)*Zbl 0815.47068

Let $E$ be a Banach space and let $C$ be a nonempty closed convex subset of $E$. Let $D$ be a subset of $C$ and let $P$ be a mapping of $C$ into $D$. Then $P$ is said to be sunny if $P(Px+t(x-Px\left)\right)=Px$ whenever $Px+t(x-Px)\in C$ for $x\in C$ and $t\ge 0$. A mapping $P$ of $C$ into $C$ is said to be a retraction if ${P}^{2}=P$. A subset $D$ of $C$ is said to be sunny nonexpansive retract on $C$ if there exists a sunny and nonexpansive retraction of $C$ onto $D$.

In the present paper the authors establish, among others, the following theorem: Let $E$ be a uniformly convex Banach space with a Fréchet differentiable norm and let $C$ be a nonempty closed convex subset of $E$. Let ${C}_{1},{C}_{2},\cdots ,{C}_{r}$ be sunny nonexpansive retracts of $C$ such that ${\bigcap}_{i=1}^{r}{C}_{i}\ne \varnothing $. Let $T$ be an operator on $C$ given by $T={\sum}_{i=1}^{r}{\alpha}_{i}{T}_{i}$, $0<{\alpha}_{i}<1$, $i=1,2,\cdots ,r$, ${\sum}_{i=1}^{r}{\alpha}_{i}=1$, such that for each $i$, ${T}_{i}=(1-{\lambda}_{i})I+{\lambda}_{i}{P}_{i}$, $0<{\lambda}_{i}<1$, where ${P}_{i}$ is a sunny nonexpansive retraction of $C$ onto ${C}_{i}$. Then $\text{Fix}\left(T\right)={\bigcap}_{i=1}^{r}{C}_{i}$ and further, for each $x\in C$, $\left\{{T}^{n}x\right\}$ converges weakly to an element of $\text{Fix}\left(T\right)$.

Finally the authors prove a common fixed point theorem for a finite commuting family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

##### MSC:

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |