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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 \(\bigcap^r_{i= 1}C_i\neq\emptyset\). 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.

MSC:

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