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Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. (English) Zbl 1090.47055
In [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043)], H.-K. Xu and R. G. Ori introduced an implicit iteration process for a finite family of nonexpansive mapping and proved the weak convergence of this process to a common fixed point of a finite family of nonexpansive maps in a Hilbert space. They posed the question: What assumptions would have to be made on the finite family of nonexpansive mappings and/or their parameters {α n } to guarantee the strong convergence of the sequence {x n } generated by their implicit iteration process? The authors prove the strong convergence of the sequence {x n } in a much more general uniformly convex Banach space under the condition that one member of the finite family of nonexpansive mappings is semi-compact or any one of the contractive assumptions of Proposition 3.4 of their paper holds.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J05Equations involving nonlinear operators (general)