In this paper we apply methods of proof mining to obtain a highly uniform effective rate of asymptotic regularity for the Ishikawa iteration associated with nonexpansive self-mappings of convex subsets of a class of uniformly convex geodesic spaces. Moreover, we show that these results are guaranteed by a combination of logical metatheorems for classical and semi-intuitionistic systems.

This paper discusses what kind of quantitative information one can extract under which circumstances from proofs of convergence statements in analysis. We show that from proofs using only a limited amount of the law-of-excluded-middle, one can extract functionals , where L is a learning procedure for a rate of convergence which succeeds after at most B-many mind changes. This -learnability provides quantitative information strictly in between a full rate of convergence and a rate of metastability in the sense of Tao (...) . In fact, it corresponds to rates of metastability of a particular simple form. Moreover, if a certain gap condition is satisfied, then B and L yield a bound on the number of possible fluctuations. We explain recent applications of proof mining to ergodic theory in terms of these results. (shrink)