Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings

Abstract

This paper establishes explicit quantitative bounds on the computation of approximate fixed points of asymptotically (quasi-) nonexpansive mappings f by means of iterative processes. Here f is a selfmapping of a convex subset C of a uniformly convex normed space X. We consider general Krasnoselski-Mann iterations with and without error terms. As a consequence of our quantitative analysis we also get new qualitative results which show that the assumption on the existence of fixed points of f can be replaced by the existence of approximate fixed points only. We explain how the existence of effective uniform bounds in this context can be inferred already a-priorily by a logical metatheorem recently proved by the first author. Our bounds were in fact found with the help of the general logical machinery behind the proof of this metatheorem. The proofs we present here are, however, completely self-contained and do not require any tools from logic.