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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 $\left\{{\alpha }_{n}\right\}$ to guarantee the strong convergence of the sequence $\left\{{x}_{n}\right\}$ generated by their implicit iteration process? The authors prove the strong convergence of the sequence $\left\{{x}_{n}\right\}$ 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:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J05 Equations involving nonlinear operators (general)