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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 $C_1,C_2,\dots, C_r$ be nonexpansive retracts of $C$ such that $\bigcap^r_{i= 1}C_i\ne\emptyset$. Assume that $T$ is a mapping on $C$ given by the formula $T= \sum^r_{i= 1}\alpha_i T_i$, where $\alpha_i\in(0, 1)$, $\sum^r_{i= 1}\alpha_i= 1$ and $T_i= (1-\lambda_i)I+ \lambda_i P_i$, where $\lambda_i\in(0, 1)$ and $P_i$ is a nonexpansive retraction of $C$ onto $C_i$ $(i= 1,2,\dots,r)$. 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 $F(T)= \bigcap^r_{i= 1}C_i$ and for every $x\in C$ the sequence $\{T^nx\}$ 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.

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