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Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces. (English) Zbl 0904.47045
The paper deals with the so-called convex feasibility problem in Banach space setting. Roughly speaking this problem is formulated as follows: Let C be a nonempty convex closed subset of a Banach space E and let C 1 ,C 2 ,,C r be nonexpansive retracts of C such that i=1 r C i . Assume that T is a mapping on C given by the formula T= i=1 r α i T i , where α i (0,1), i=1 r α i =1 and T i =(1-λ i )I+λ i P i , where λ i (0,1) and P i is a nonexpansive retraction of C onto C i (i=1,2,,r). One can find assumptions concerning the space E which guarantee that the set F(T) of fixed points of the mapping T can be represented as F(T)= i=1 r C i and for every xC the sequence {T n x} converges weakly to an element of F(T). The authors prove that if E is a uniformly convex Banach space with a Fréchet differentiable norm (or a reflexive and strictly convex Banach space satisfying the Opial condition) then the convex feasibility problem has a positive solution. Apart from that the problem of finding a common fixed point for a finite commuting family of nonexpansive mappings in a strictly convex and reflexive Banach space is also considered.

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