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Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings. (English) Zbl 1132.65048

Summary: We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space \(E\). We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case \(E\) is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial’s condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

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