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Image recovery by convex combinations of sunny nonexpansive retractions. (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 $P(Px+ t(x- Px))= Px$ whenever $Px+ t(x- Px)\in C$ for $x\in C$ and $t\geq 0$. A mapping $P$ of $C$ into $C$ is said to be a retraction if $P\sp 2=P$. 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 $C\sb 1, C\sb 2,\dots, C\sb r$ be sunny nonexpansive retracts of $C$ such that $\bigcap\sb{i=1}\sp r C\sb i \ne \emptyset$. Let $T$ be an operator on $C$ given by $T= \sum\sb{i=1}\sp r \alpha\sb i T\sb i$, $0<\alpha\sb i< 1$, $i=1,2, \dots,r$, $\sum\sb{i=1}\sp r \alpha\sb i=1$, such that for each $i$, $T\sb i= (1- \lambda\sb i) I+ \lambda\sb i P\sb i$, $0<\lambda\sb i <1$, where $P\sb i$ is a sunny nonexpansive retraction of $C$ onto $C\sb i$. Then $\text{Fix} (T)= \bigcap\sb{i=1}\sp r C\sb i$ and further, for each $x\in C$, $\{T\sp n x\}$ converges weakly to an element of $\text{Fix} (T)$. 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.

47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces