The Halpern-Mann iteration in CAT(0) spaces.

Abstract

Complete CAT(0) spaces, also known as Hadamard spaces, are a non-linear generalization of Hilbert spaces. Benefiting from ideas and tools from the proof mining program it was shown by Dinis and Pinto the strong convergence of an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations in the general context of CAT(0) spaces. At the same time, the logical tools used allowed to obtain quantitative information in the form of rates of asymptotic regularity and rates of metastability (in the sense of T. Tao). If one restricts oneself to Hilbert spaces, the proof follows some standard arguments. However, to obtain the proof in the more general context of CAT(0) spaces the use of logical tools, and in particular the technique introduced by Ferreira et al., seem to be necessary. In this talk I will explain the role of the logical tools in obtaining this result.