Abstract not found

Abstract not found

Abstract not found

this paper is to introduce and analyze a common algorithmic framework encompassing and extending the above iterative methods. The algorithm under consideration is the following inexact, Mann-like generalization of (5) xn+1 = xn + n Tnxn + e n xn where e n 2 H; 0 < n < 2; and Tn 2 T : (10) Here, e n stands for the error made in the computation of Tnxn ; incorporating such errors provides a more realistic model of the actual implementation of the algorithm. Throughout, the convex combinations in (10) are de ned as xn = n;j x j ; (11) 3 where ( n;j ) n;j0 are the entries of an in nite lower triangular row stochastic matrix A, i.e., > > (8j 2 N) n;j 0 (8j 2 N) j > n ) n;j = 0 j=0 n;j = 1; (12) which satis es the regularity condition (8j 2 N) lim n!+1 n;j = 0: (13) Our analysis will not rely on the segmenting condition (7) and will provide convergence results for the inexact, extended Mann iterations (10) for a wide range of averaging schemes

Abstract. In this paper, we examine the stability of Kirk-Ishikawa and Kirk-Mann iteration processes for nonexpansive and quasi-nonexpansive operators in uniformly convex Banach space. To the best of our knowledge, apart from the results of Olatinwo [19], stability of fixed point iteration processes has not been investigated in uniformly convex Banach space. Our results generalize, extend and improve some of the results of Harder and Hicks [11], Rhoades [26, 27], Osilike [23], Berinde [2, 3] as well as Imoru and Olatinwo [12]. Key words: uniformly convex Banach space; Ishikawa iteration

In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in CAT(0)-spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch’s theorem obtained by Kohlenbach using methods from mathematical logic (so-called “proof mining”). Keywords: MSC: Proof mining, metric fixed point theory, nonexpansive functions,

We propose the class of uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity (UCW-hyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCW-hyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT(0)-spaces. Furthermore, we apply proof mining techniques to get effective rates of asymptotic regularity for Ishikawa iterations of nonexpansive self-mappings of closed convex subsets in UCW-hyperbolic spaces. These effective results are new even for uniformly convex Banach spaces. 1

This paper is part of the general project of proof mining, developed by Kohlenbach. By ”proof mining ” we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, R-trees and uniformly convex hyperbolic spaces. Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee a-priori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixed-point theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex hyperbolic spaces obtained in a previous paper is an instance of one of our metatheorems. Keywords: MSC: Proof mining, hyeprbolic spaces, R-trees, asymptotic regularity,

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 selfmappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of proof mining, which is concerned with the extraction of effective uniform bounds from (prima-facie) ineffective proofs. 1