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 Gödel’s system [Formula: see text] 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 [Formula: see text]. 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.

PROOF MINING IN Lp SPACES

We obtain an equivalent implicit characterization of Lp Banach spaces that is amenable to a logical treatment. Using that, we obtain an axiomatization for such spaces into a higher order logical system, the kind of which is used in proof mining, a research program that aims to obtain the hidden computational content of mathematical proofs using tools from mathematical logic. As an aside, we obtain a concrete way of formalizing Lp spaces in positive-bounded logic. The axiomatization is followed by a corresponding metatheorem in the style of proof mining. We illustrate its use with the derivation for this class of spaces of the standard modulus of uniform convexity.

Effective results on nonlinear ergodic averages in CAT spaces

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\unicode[STIX]{x1D705})$ spaces (with $\unicode[STIX]{x1D705}>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.