We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann 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 well-known 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 ...

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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, Hölder-Lipschitz, 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.

Let (x n ) and (x n ) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (y n ) with better convergence properties than (x n ) and (x n ). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.

The Lanczos method for solving Ax = b consists in constructing the sequence of vectors x k such that r k = b \Gamma Ax k = P k (A)r 0 where P k is the orthogonal polynomial of degree at most k with respect to the linear functional c whose moments are c(¸ i ) = c i = (y

This paper is a case study in proof mining applied to non-effective 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 so-called Krasnoselski-Mann 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 non-uniqueness 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 non-effective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...

In a repeated-interaction public goods economy, dynamic behavior may affect the efficiency of various mechanisms thought to be efficient in one-shot 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 Vickrey-Clarke-Groves mechanism help to draw out this result. The simplicity of the model makes it useful in predicting dynamic stability of other mechanisms.

The aim of this paper is to study the multiple hybrid procedure which produces a better residual vector from those given by several iterative methods used simultaneously for solving a system of linear equations. A particular case leads to new semi-iterative methods where the residual is minimized instead of the error. Discussions about several related topics such as projection, extrapolation, best approximation and biorthogonalization are also included and generalizations of the Lanczos method are discussed. Numerical examples illustrate the purpose.

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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, 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