The Alternating Halpern-Mann Iteration for Families of Maps

Abstract

We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.