41 research outputs found
Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings
In this paper we obtain new effective results on the Halpern iterations of
nonexpansive mappings using methods from mathematical logic or, more
specifically, proof-theoretic techniques. We give effective rates of asymptotic
regularity for the Halpern iterations of nonexpansive self-mappings of nonempty
convex sets in normed spaces. The paper presents another case study in the
project of {\em proof mining}, which is concerned with the extraction of
effective uniform bounds from (prima-facie) ineffective proofs.Comment: in C.S. Calude, G. Stefanescu, and M. Zimand (eds.), Combinatorics
and Related Areas. A Collection of Papers in Honour of the 65th Birthday of
Ioan Tomesc
Effective results on compositions of nonexpansive mappings
This paper provides uniform bounds on the asymptotic regularity for
iterations associated to a finite family of nonexpansive mappings. We obtain
our quantitative results in the setting of -convex spaces, a class
of geodesic spaces which generalizes metric spaces with a convex geodesic
bicombing
Proof mining in metric fixed point theory and ergodic theory
In this survey we present some recent applications of proof mining to the
fixed point theory of (asymptotically) nonexpansive mappings and to the
metastability (in the sense of Terence Tao) of ergodic averages in uniformly
convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page
Alternative iterative methods for nonexpansive mappings, rates of convergence and application
Alternative iterative methods for a nonexpansive mapping in a Banach space
are proposed and proved to be convergent to a common solution to a fixed point
problem and a variational inequality. We give rates of asymptotic regularity
for such iterations using proof-theoretic techniques. Some applications of the
convergence results are presented
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Alternative iterative methods for nonexpansive mappings, rates of convergence and applications
Alternative iterative methods for a nonexpansive mapping in a Banach space are
proposed and proved to be convergent to a common solution to a fixed point problem and
a variational inequality. We give rates of asymptotic regularity for such iterations using
proof-theoretic techniques. Some applications of the convergence results are presented
Linear rates of asymptotic regularity for Halpern-type iterations
In this note we apply a lemma due to Sabach and Shtern to compute linear
rates of asymptotic regularity for Halpern-type nonlinear iterations studied in
optimization and nonlinear analysis
Effective metastability of Halpern iterates in CAT(0) spaces
This paper provides an effective uniform rate of metastability (in the sense
of Tao) on the strong convergence of Halpern iterations of nonexpansive
mappings in CAT(0) spaces. The extraction of this rate from an ineffective
proof due to Saejung is an instance of the general proof mining program which
uses tools from mathematical logic to uncover hidden computational content from
proofs. This methodology is applied here for the first time to a proof that
uses Banach limits and hence makes a substantial reference to the axiom of
choice.Comment: some typos correcte