## 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 $$P^ 2=P$$. 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 $$C_ 1, C_ 2,\dots, C_ r$$ be sunny nonexpansive retracts of $$C$$ such that $$\bigcap_{i=1}^ r C_ i \neq \emptyset$$. Let $$T$$ be an operator on $$C$$ given by $$T= \sum_{i=1}^ r \alpha_ i T_ i$$, $$0<\alpha_ i< 1$$, $$i=1,2, \dots,r$$, $$\sum_{i=1}^ r \alpha_ i=1$$, 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 $$\text{Fix} (T)= \bigcap_{i=1}^ r C_ i$$ and further, for each $$x\in C$$, $$\{T^ n x\}$$ 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.

### MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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