Recently, Cheval and Leustean computed linear rates of asymptotic regularity for a viscosity version of Halpern-type iterations generated by a nonexpansive self-mapping of a metric space assuming that it has a fixed point. In the present paper, we show that their result is valid without this assumption. It is enough either to assume the existence of a bounded orbit of the nonexpansive mapping or the existence of a bounded sequence of approximate fixed points of the mapping.
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Alexander J. Zaslavski (2024) studied this question.
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