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 differentproof 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.

On computational properties of Cauchy problems generated by accretive operators

“…Methodologically, the results of this work are based on the general approach and methods of the 'proof mining' program, a discipline of mathematical logic which aims at the extraction of quantitative information from prima facie nonconstructive proofs by logical transformations (see [23] for a book treatment and [26] for a recent survey). In that vein, this work can in particular be viewed as a new case study in this program and is further strongly related to the only other previous foray of proof mining into the theory of partial differential equations and abstract Cauchy problems presented by Kohlenbach and Koutsoukou-Argyraki in [27] (as well as to the only two other previous considerations on nonexpansive semigroups presented in [28,34]).…”

On computational properties of Cauchy problems generated by accretive operators

“…Methodologically, the results of this work are based on the general approach and methods of the "proof mining" program, a discipline of mathematical logic which aims at the extraction of quantitative information from prima facie nonconstructive proofs by logical transformations (see [21] for a book treatment and [24] for a recent survey). In that vein, this work can in particular be viewed as a new case study in this program and is further strongly related to the only other previous foray of proof mining into the theory of partial differential equations and abstract Cauchy problems presented by Kohlenbach and Koutsoukou-Argyraki in [25] (as well as to the only two other previous considerations on nonexpansive semigroups presented in [26,32]).…”

Formalising Mathematics – in Praxis; A Mathematician’s First Experiences with Isabelle/HOL and the Why and How of Getting Started

“…As a pure mathematician with some background in logic and proof theory, my own interest in the topic was initially driven not only by a fascination for the emerging culture of re-imagining mathematical practice in the light of new AI developments but also by philosophical questions on the nature of mathematical proofs, e.g. when encountering in my own research work different proofs of similar statements giving completely different computational content [29,31].…”

“…In particular, regarding my mathematics background, my PhD research [32] was pen-and-paper work and involved applications of proof theory to mathematics (mainly in nonlinear analysis) [28][29][30][31]. I have been working within Ulrich Kohlenbach's proof mining programme [25][26][27] that involves pen-and-paper extraction of constructive/quantitative information from proofs in the form of computable bounds, which requires a logical analysis of a proof and rewriting it to make the logical form of all the statements involved explicit via revealing the hidden quantifiers.…”

“…In the aforementioned proof mining papers [29,31] it turns out that different proofs of similar statements (about the fixed points of nonexpansive semigroups) give completely different computational content, which in turn, provokes the question: What do we mean by "better" computational content?…”