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

Angeliki Koutsoukou-Argyraki

doi:10.4115/jla.2018.10.7

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.

Highlights

Proof mining is a research program in applied proof theory originally initiated by Georg Kreisel in the 1950’s under the name unwinding of proofs, after he suggested a shift of focus for the application of proof interpretations: from producing relative consistency proofs for foundational purposes to a tool for extracting constructive information from actual mathematical proofs
This information is “hidden” behind an implicit use of quantifiers in the proof, and its extraction is guaranteed by certain logical metatheorems, see for example Kohlenbach [9], provided that the statement proved is of a certain logical form and the assumptions and general mathematical setup fit a specific formal framework
[6], proof mining has been applied by Kohlenbach and his collaborators to various fields of Mathematics, including approximation theory, ergodic theory, fixed point theory, nonlinear analysis and recently PDE theory

Summary

Introduction

Proof mining is a research program in applied proof theory originally initiated by Georg Kreisel in the 1950’s under the name unwinding of proofs (see [16] or [17]), after he suggested a shift of focus for the application of proof interpretations: from producing relative consistency proofs for foundational purposes to a tool for extracting constructive information from actual mathematical proofs. It is very interesting that the bound obtained here (Section 3) is completely different to the bound obtained in another recent work [13] by Kohlenbach and the author The latter had been derived by proof mining on a proof of a statement again by Suzuki in [21] concerning, again, the common fixed points of {T(t) : t ≥ 0} that is a generalization of the corresponding statement in [20]. In the last section we give a corollary to our result for the case of uniformly convex C: we apply our result to extract a rate of asymptotic regularity for {T(t) : t ≥ 0} with respect to a classical iteration schema introduced by Kuhfittig in [18] This is achieved by making use of a theorem by Khan and Kohlenbach [5] which was derived via proof mining on the proof of a result by Kuhfittig [18]

Preliminaries

Asymptotic Regularity