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.