arXiv subjects

Daniel Körnlein

Publications and source records attributed to Daniel Körnlein.

3 recordsLinked to original sources

arXiv2016

Quantitative results for Bruck iterations of demicontinuous pseudocontractions

Our first result is a rate of metastability in the sense of Tao for Bruck's iteration scheme for demicontinuous pseudocontractions in Hilbert space, extracted from Bruck's original proof. This result generalizes earlier work in the ongoing program of proof mining from Lipschitzian to demicontinuous pseudocontractions. Our second main result is a metastable version of asymptotic regularity under the additional assumption that the underlying operator is norm-to-norm uniformly continuous on bounded subsets. These results (and their intermediate versions given in this paper) provide a thorough quantitative analysis of Bruck's iteration scheme for pseudocontractions in Hilbert space.

math.LO↗

arXiv2016

Quantitative Strong Convergence for the Hybrid Steepest Descent Method

We provide new complexity information for the convergence of the Hybrid Steepest Descent Method for solving the Variational Inequality Problem for a strict contraction on Hilbert space over a closed convex set C given either as the fixed point set of a single nonexpansive mapping or the intersection of the fixed point sets of a finite family of nonexpansive mappings. More precisely, we give metastability rates in the sense of Tao for those cases. The results in this paper were extracted from a proof due to Yamada using proof-mining techniques, and provide a thorough quantitative analysis of the Hybrid Steepest Descent Method.

math.LO↗

arXiv2013

Quantitative results for Halpern iterations of nonexpansive mappings

We give a rate of metastability for Halpern's iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due to Xu. In Hilbert space, the latter is known, so we get an explicit rate of metastability. We also extract a rate of asymptotic regularity for general normed spaces. Such rates have already been extracted by Kohlenbach and Leuştean for different and incomparable conditions on $(λ_n)$. The proof analyzed in this paper is also more effective than the proofs treated by Kohlenbach and Leuştean in that it does not use Banach limits or weak compactness, which makes the extraction particularly efficient. Moreover, we also give an equivalent axiomatization of uniformly smooth Banach spaces. This paper is part of an ongoing case study of proof mining in nonlinear fixed point theory.

math.FA↗