The finitary content of sunny nonexpansive retractions
Abstract
We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably total functions as firstorder arithmetic and therefore one may interpret the resulting proof in Gödel's system $T$ of highertype functionals. The witness so obtained is then majorized (in the sense of Howard) in order to produce the final bound, which is shown to be definable in the subsystem $T_1$. This piece of information is further used to obtain rates of metastability to results which were previously only analyzed from the point of view of proof mining in the context of Hilbert spaces, i.e. the convergence of the iterative schemas of Halpern and Bruck.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 DOI:
 10.48550/arXiv.1812.04940
 arXiv:
 arXiv:1812.04940
 Bibcode:
 2018arXiv181204940K
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Logic;
 47H06;
 47H09;
 47H10;
 03F10