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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 \(\{ \alpha _{n}\} \) 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:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)

Citations:

Zbl 0999.47043
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Full Text: DOI

References:

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[8] Zhou, Y.; Chang, S.S., Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. funct. anal. appl., 23, 911-921, (2002) · Zbl 1041.47048
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