Effective results on nonlinear ergodic averages in CAT spaces

Abstract

In this paper we apply proof mining techniques to compute, in the setting of CAT(κ) spaces (with κ > 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.MSC: 47H25, 03F10, 47J25, 47H09.

Effective results on compositions of nonexpansive mappings

“…More precisely, we introduce (r, δ)-convex spaces, a class of metric spaces which also includes Busemann spaces (and, hence, CAT(0) spaces), hyperconvex spaces, CAT(κ) spaces with κ > 0, as well as the so-called W -hyperbolic spaces (see [11]). Consequently, even when N = 1 and so (1) reduces in fact to the Halpern iteration, our results generalize ones obtained previously by the authors for CAT(κ) spaces with κ > 0 [17] and by the first author for normed [15] or W -hyperbolic spaces [16].…”

“…rates of convergence of ( x n − T x n ) towards 0. For the Halpern iteration this was done in a series of papers [15,16,13,14,17], corresponding to different classes of spaces. For the iteration given by (3), such rates were obtained in [6].…”

Quantitative results for Halpern iterations of nonexpansive mappings

“…Apart from rates of convergence and divergence for the conditions (i) to (iii), it only depends on an upper bound on the distance of the starting point from some fixed point of S. In the case α n = 1/n + 1, Kohlenbach [7] also improved the exponential rate of asymptotic regularity to a quadratic one. Moreover, these results were also generalised to CAT(0) spaces [8] and CAT(κ) spaces [14]. Closely related to Wittmann's result is the following …”

Effective asymptotic regularity for one-parameter nonexpansive semigroups

“…Within the past 15 years, proof mining has been applied by the first author and his collaborators to various fields of mathematics, including approximation theory, ergodic theory, fixed point theory, nonlinear analysis, and (recently) PDE theory (see e.g. [10,9,11,13]). …”

“…For instance, in [15] the following nonexpansive semigroup is studied (referring to [14] where it is attributed to G.F. Webb):…”