### New effective bounds for the approximate common fixed points and asymptotic regularity of nonexpansive semigroups

#### Abstract

We give an explicit, computable and uniform bound for the computation of approximate common fixed points of one-parameter nonexpansive semigroups on a subset $C$ of a Banach space, by proof mining on a proof by Suzuki. The bound obtained here is different to the bound obtained in a very recent work by Kohlenbach and the author which had been derived by proof mining on the -completely different- proof of a generalized version of the particular theorem by Suzuki. We give an adaptation of a logical metatheorem by Gerhardy and Kohlenbach for the given mathematical context, illustrating how the extractability of a computable bound is guaranteed. For uniformly convex $C$, as a corollary to our result we moreover give a computable rate of asymptotic regularity with respect to Kuhfittig's classical iteration schema, by applying a theorem by Khan and Kohlenbach.

#### Full Text:

7. [PDF]DOI: https://doi.org/10.4115/jla.2018.10.7

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Journal of Logic and Analysis ISSN: 1759-9008