381 research outputs found
Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces
We further study averaged and firmly nonexpansive mappings in the setting of
geodesic spaces with a main focus on the asymptotic behavior of their Picard
iterates. We use methods of proof mining to obtain an explicit quantitative
version of a generalization to geodesic spaces of result on the asymptotic
behavior of Picard iterates for firmly nonexpansive mappings proved by Reich
and Shafrir. From this result we obtain effective uniform bounds on the
asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive
effective rates of asymptotic regularity for sequences generated by two
algorithms used in the study of the convex feasibility problem in a nonlinear
setting
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
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 page
The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings
In this paper we provide a unified treatment of some convex minimization
problems, which allows for a better understanding and, in some cases,
improvement of results in this direction proved recently in spaces of curvature
bounded above. For this purpose, we analyze the asymptotic behavior of
compositions of finitely many firmly nonexpansive mappings in the setting of
-uniformly convex geodesic spaces focusing on asymptotic regularity and
convergence results
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