Abstract:
This paper establishes explicit quantitative bounds on the
computation of approximate fixed points of asymptotically (quasi-)
nonexpansive mappings 
by means of iterative processes. Here 
is
a selfmapping of a convex subset 
of a uniformly convex normed
space 
. 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 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 selfcontained and do not
require any tools from logic.
by means of iterative processes. Here is
a selfmapping of a convex subset of a uniformly convex normed
space . 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 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 selfcontained and do not
require any tools from logic.Available as PostScript, PDF, DVI.