Sharp convergence rates for averaged nonexpansive maps

DOI10.1007/S11856-018-1723-ZarXiv1606.05300WikidataQ129617370 ScholiaQ129617370MaRDI QIDQ6274702

Roberto Cominetti, Mario Bravo

Publication date: 16 June 2016

Abstract: We establish sharp estimates for the convergence rate of the Kranosel'skiv{i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of Mathematics 199(2), 757-772, 2014) with constant $1/sqrtpi$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in [3] (Proceedings of the 2nd International Conference on Fixed Point Theory and Applications, World Scientific Press, London, 27-66, 1992) which we reinterpret in terms of a nested family of optimal transport problems. We show that these bounds are tight by building a nonexpansive map $T:[0,1{]}^{mathbbN}o[0,1{]}^{mathbbN}$ that attains them with equality, settling the main conjecture in [3]. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant $1/sqrtpi$.

Mathematics Subject Classification ID

Equations and inequalities involving nonlinear operators (47Jxx)

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