Laurentiu Leustean
Romanian Academy
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Featured researches published by Laurentiu Leustean.
Abstract and Applied Analysis | 2003
Ulrich Kohlenbach; Laurentiu Leustean
In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets C in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich, and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions. In this paper, we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W. A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.
Journal of the European Mathematical Society | 2010
Ulrich Kohlenbach; Laurentiu Leustean
This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.
Transactions of the American Mathematical Society | 2014
David Ariza Ruiz; Laurentiu Leustean; Genaro López Acedo
Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard iterates of these mappings in different classes of geodesic spaces, such as (uniformly convex) W -hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic regularity for the Picard iterations. MSC: Primary: 47H09, 47H10, 53C22; Secondary: 03F10, 47H05, 90C25, 52A41.
Philosophical Transactions of the Royal Society A | 2012
Ulrich Kohlenbach; Laurentiu Leustean
This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.
Communications in Contemporary Mathematics | 2018
Ulrich Kohlenbach; Laurentiu Leustean; Adriana Nicolae
We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called W-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.
Ergodic Theory and Dynamical Systems | 2016
Laurentiu Leustean; Adriana Nicolae
In this paper we apply proof mining techniques to compute, in the setting of CAT
Annals of Pure and Applied Logic | 2014
Laurentiu Leustean
(\kappa)
Numerical Functional Analysis and Optimization | 2015
Daniel Ivan; Laurentiu Leustean
spaces (with
Journal of Convex Analysis | 2013
Vittorio Colao; Genaro López; Laurentiu Leustean; Victoria Martín-Márquez
\kappa >0
Archiv der Mathematik | 2015
Laurentiu Leustean; Adriana Nicolae
), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.