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The general iteration method for nonexpansive mappings on a Banach
space is considered. Under some assumption of fast enough convergence on the
sequence of (“almost” nonexpansive) perturbed iteration mappings, if the basic
method is τ−convergent for a suitable topology τ weaker than the norm topology,
then the perturbed method is also τ−convergent. Application is presented to the
gradient-prox method for monotone inclusions in Hilbert spaces.
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