Results 1  10
of
198
General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
Abstract

Cited by 45 (26 self)
 Add to MetaCart
(Show Context)
In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
A quantitative version of a theorem due to BorweinReichShafrir
 Numerical Functional Analysis and Optimization
, 2000
"... We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general KrasnoselskiMann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerni ..."
Abstract

Cited by 25 (14 self)
 Add to MetaCart
(Show Context)
We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general KrasnoselskiMann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by wellknown results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error #, an upper bound on the diameter of C and some very general information on the sequence of scalars # k used in the iteration. Only in the special situation, where # k := # is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was ...
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 ..."
Abstract

Cited by 24 (4 self)
 Add to MetaCart
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”).
Learning Dynamics In Mechanism Design: An Experimental Comparison Of Public Goods Mechanisms
, 2003
"... In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
(Show Context)
In a repeatedinteraction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in oneshot games. Inspired by results obtained in previous experiments, the current paper proposes a simple best response model in which players' beliefs are functions of previous strategy profiles. The predictions of the model are found to be highly consistent with new experimental data from five mechanisms with various types of equilibria. Interesting properties of a 2parameter VickreyClarkeGroves mechanism help to draw out this result. The simplicity of the model makes it useful in predicting dynamic stability of other mechanisms.
Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings
 in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications
, 2009
"... Abstract. In this article, we prove strong and weak convergence theorems for finding a common element of the set of solutions for an equilibrium problen $td $ the set of fixed points of arelatively nonexpansive mapping in aBanach space. Next, we prove two strong convergence theorems for finding acon ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In this article, we prove strong and weak convergence theorems for finding a common element of the set of solutions for an equilibrium problen $td $ the set of fixed points of arelatively nonexpansive mapping in aBanach space. Next, we prove two strong convergence theorems for finding aconmon element of the zero point set of amaximal monotone operator and the fixed point set of arelatively nonexpaoive mapping in aBanai spaoe by using the normal hybrid method and anew hybrid method called the $shr\dot{i}ingproj\infty tion $ method. $E\backslash lrther $ , we obtain a $n\infty essary $ and sufficient conition for the existence of $8olutions $ of the euquihbrium problem by using the metric resolvents. Finally, we prove a $\epsilon trong $ convergence thmrem for findin$g $ asolution of an equihbrium problem in aBanach space by using the $shr\dot{i}$king projection method. 1 lntroduction Let $E $ be a real Banach space and let $E$ “ be a dual space of $E$. Let $C $ be a closed convex subset of $E $ and let $f $ be a bifunction from $CxC $ to $R $ , where $R $ is the set of real numbers. The equilibrium problem is formulated as follows: Find $\hat{x}\in C $ such that $f(\hat{x},y)\geq 0 $ for all $y\in C $.
A Mann iterative regularization method for elliptic Cauchy problems
 Numer. Funct. Anal. Optim
, 2001
"... We investigate the Cauchy problem for linear elliptic operators with C1–coefficients at a regular set R2, which is a classical example of an illposed problem. The Cauchy data are given at the manifold @ and our goal is to reconstruct the trace of theH1ðÞ solution of an elliptic equation at @= ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
We investigate the Cauchy problem for linear elliptic operators with C1–coefficients at a regular set R2, which is a classical example of an illposed problem. The Cauchy data are given at the manifold @ and our goal is to reconstruct the trace of theH1ðÞ solution of an elliptic equation at @=. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed byMaz’ya et al., who proposed a method based on solving successive wellposed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically. 1.
Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces
 Fixed Point Theory and Applications
, 2005
"... In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1. ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1.
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...