Quantitative results on the Ishikawa iteration of Lipschitz pseudo-contractions

TL;DR

This paper applies proof mining techniques to derive explicit rates of metastability for the Ishikawa iteration of Lipschitz pseudo-contractions in Hilbert spaces, revealing hidden quantitative information.

Contribution

It provides the first explicit uniform rates of metastability for this iteration, extending previous qualitative convergence results using logical tools.

Findings

Derived explicit rates of metastability for the iteration

Extended proof mining methods to Lipschitz pseudo-contractions

Enhanced understanding of convergence behavior in Hilbert spaces

Abstract

We compute uniform rates of metastability for the Ishikawa iteration of a Lipschitz pseudo-contractive self-mapping of a compact convex subset of a Hilbert space. This extraction is an instance of the proof mining program that aims to apply tools from mathematical logic in order to extract the hidden quantitative content of mathematical proofs. We prove our main result by applying methods developed by Kohlenbach, the first author and Nicolae for obtaining quantitative versions of strong convergence results for generalized Fej\'er monotone sequences in compact subsets of metric spaces.