# The finitary content of sunny nonexpansive retractions

## TL;DR

This paper employs proof mining techniques to extract explicit uniform rates of metastability for the convergence of approximants to fixed points of pseudocontractive mappings in certain Banach spaces, extending classical results.

## Contribution

It introduces a novel proof mining approach to derive explicit convergence rates in Banach spaces with specific geometric properties, utilizing the existence of a modulus of uniqueness.

## Key findings

- Derived a uniform rate of metastability for fixed point approximations
- Extended proof mining techniques to Banach spaces with uniform convexity and smoothness
- Produced explicit bounds interpretable in higher-type systems

## 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 first-order arithmetic and therefore one may interpret the resulting proof in G\"odel's system $T$ of higher-type 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.