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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 ..."
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Cited by 31 (18 self)
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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, 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 ..."
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Cited by 19 (13 self)
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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 ...
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 ..."
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Cited by 11 (10 self)
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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...
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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Cited by 8 (0 self)
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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”).
2 1 General introduction
"... This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective unif ..."
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This paper (and its companion [29]) is another case study in the project of ‘proof mining ’ 1 in analysis by which we mean the logical analysis of mathematical proofs (typically using noneffective analytical tools) with the aim of extracting new numerically relevant information (e.g. effective uniform bounds or algorithms etc.) hidden in the proofs. 2 Many problems in numerical (functional) analysis can be seen as instances of the following general task: construct a solution x of an equation A(x): ≡ (F(x) = 0), where x is an element of some Polish (i.e. complete separable metric) space (typically with additional structure) and F: X → IR (usually F will depend on certain parameters a which again belong to Polish spaces). Quite often the construction of such a solution is obtained in two steps: 1) One shows how to construct (uniformly in the parameters of A) approximate solutions (sometimes called ‘εsolutions’) xε ∈ X for an εversion of the original equation Aε(x): ≡ ( F(x)  < ε). 2) Exploiting compactness conditions on X one concludes that either (x 1 n)n∈IN itself or some subsequence of it converges to a solution of A(x). The first step usually is constructive. However, the noneffectivity of the second step in many cases prevents one from being able to compute a solution x of A effectively within a prescribed error 1 k, i.e. to compute a function n(k) such that dX(xn(k), x) < 1 k. In many cases X: = K is compact and x is uniquely determined. Then (xn) itself converges to x so that no subsequence needs to be selected. However, the problem of how to get apriori bounds (in particular not depending on x itself) on the rate of convergence of that sequence remains. In numerical analysis, often such rates are not provided (due to the ineffectivity of the proof of the uniqueness of x). 3 In a series of papers we have demonstrated the applicability of proof theoretic techniques to extract socalled uniform moduli of uniqueness (which generalize 1 The term ‘proof mining ’ (instead of G. Kreisel’s ‘unwinding of proofs’) for the activity of extracting additional information hidden in given proofs using proof theoretic tools was suggested to the author by Professor Dana Scott. 2 For a different case study in analysis in the context of best approximation theory see [21],[22]. For other kinds of logical analyses of specific proofs see [33] and [36].