The finitary content of sunny nonexpansive retractions

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 Godel'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 hound, 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.

Automated summary

Using proof mining techniques, we extract a uniform rate of metastability for strong convergence of approximants to fixed points of pseudocontractive mappings in uniformly convex and smooth Banach spaces, a restricted form of the classical Reich's result. The existence of a unique modulus specific to uniformly convex Banach spaces and the arithmetization of limit superior allow us to prove metastable convergence in a system with total functions equivalent to first-order arithmetic, interpretable in Godel's system T of higher-type functionals. The obtained witness is then majorized to produce a final hound definable in subsystem T-1, leading to obtaining rates of metastability in results previously only analyzed from the perspective of proof mining in the context of Hilbert spaces.