On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

Abstract

We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(ĸ)-spaces (ĸ > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.