10.2168/LMCS-11(4:20)2015
Neumann, Eike
Eike
Neumann
Computational Problems in Metric Fixed Point Theory and their Weihrauch
Degrees
episciences.org
2015
Mathematics - Logic
Computer Science - Logic in Computer Science
contact@episciences.org
episciences.org
2013-12-08T00:00:00+01:00
2016-11-21T15:20:51+01:00
2015-12-29
eng
Journal article
https://lmcs.episciences.org/1621
arXiv:1506.05127
1860-5974
PDF
1
Logical Methods in Computer Science ; Volume 11, Issue 4 ; 1860-5974
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.
Comment: 44 pages