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13
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
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A quadratic rate of asymptotic regularity for CAT(0)spaces
, 2005
"... In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann 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 hy ..."
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In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann 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 (socalled “proof mining”).
Generalized Mann iterates for constructing fixed points in Hilbert spaces
"... 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, Mannlike 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, ..."
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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, Mannlike 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
Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces, Nonlinear Anal
"... This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖T xn −xn ‖ → 0 due to Chidume and Zegeye in 2004 where (xn) is a certain pertubated KrasnoselskiMann iteration schema for Lipschitz pseudocontractive selfmappings T of closed and convex subsets of ..."
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This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖T xn −xn ‖ → 0 due to Chidume and Zegeye in 2004 where (xn) is a certain pertubated KrasnoselskiMann iteration schema for Lipschitz pseudocontractive selfmappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved. 1
SOME STABILITY RESULTS FOR NONEXPANSIVE AND QUASINONEXPANSIVE OPERATORS IN UNIFORMLY CONVEX BANACH SPACE USING TWO NEW ITERATIVE PROCESSES OF KIRKTYPE
"... Abstract. In this paper, we examine the stability of KirkIshikawa and KirkMann iteration processes for nonexpansive and quasinonexpansive 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 ..."
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Abstract. In this paper, we examine the stability of KirkIshikawa and KirkMann iteration processes for nonexpansive and quasinonexpansive 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
Nonexpansive iterations in uniformly convex
, 810
"... We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic spaces are a natural generalization both of uniformly convex normed spaces and ..."
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We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic 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 selfmappings of closed convex subsets in UCWhyperbolic spaces. These effective results are new even for uniformly convex Banach spaces. 1
Proof mining in Rtrees and hyperbolic spaces
, 801
"... 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 ..."
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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, Rtrees and uniformly convex hyperbolic spaces. Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee apriori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixedpoint theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the KrasnoselskiMann 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, Rtrees, asymptotic regularity,