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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)C for xC and t0. 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 ,,C r be sunny nonexpansive retracts of C such that i=1 r C i . Let T be an operator on C given by T= i=1 r α i T i , 0<α i <1, i=1,2,,r, i=1 r α i =1, such that for each i, T i =(1-λ i )I+λ i P i , 0<λ i <1, where P i is a sunny nonexpansive retraction of C onto C i . Then Fix(T)= i=1 r C i and further, for each xC, {T n x} converges weakly to an element of 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:
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces