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Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach spaces. (English) Zbl 1109.47056

The paper presents a strong convergence theorem for an iterative scheme of Halpern type used to approximate common fixed points of a finite family of nonexpansive mappings \(T_1,T_2,\dots,T_n\) which are defined on a uniformly smooth Banach space with a weakly sequentially continuous duality mapping. To obtain the result, it is explicitly assumed that the common fixed point set \(F=\bigcap \limits_{i=1}^{n}F(T_i)\) is nonempty and all circular compositions of \(T_1,T_2,\dots,T_n\) have the same fixed point set \(F\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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