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We characterize the family of iterated nonexpansive mappings that are stable under every renorming. The family of iterated nonexpansive mappings contains the family of nonexpansive mappings, it also contains quasi-nonexpansive and Suzuki's (C)-type mappings with fixed points, among others. We also give the corresponding characterizations for quasi-nonexpansive and some Suzuki's (C)-type mappings with fixed points.
Pérez-García et al. (Tue,) studied this question.