Some logical metatheorems with applications in functional analysis

By Ulrich Kohlenbach. 2003

362 citations recorded by the source corpus.

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here 'uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ac25236d6a694a24d1f3b9295295c662fdd199a3

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Citations in the corpus

Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces (2007, methodology): “Precise definitions for all this can be found in [13, 4].”
Nonexpansive iterations in uniformly convex $W$-hyperbolic spaces (2008, methodology): “2 UCW -hyperbolic spaces We work in the setting of hyperbolic spaces as introduced by Kohlenbach [12].”
LOCAL STABILITY OF ERGODIC AVERAGES (2007, methodology): “Our extractions of bounds can be viewed as applications of a body of proof theoretic results that fall under the heading “proof mining” (see, for example, [19, 20, 21]).”
PROOF-THEORETIC METHODS IN NONLINEAR ANALYSIS (2019, background): “For the case of bounded hyperbolic spaces, see [32].”
Firmly nonexpansive mappings in classes of geodesic spaces (2012, methodology): “MR2445721 (2009k:03003) [26] U. Kohlenbach and L. Leuştean, Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, J. Eur.”
A quadratic rate of asymptotic regularity for CAT(0)-spaces (2005, methodology): “By a logical analysis of the proof of a theorem due to Borwein/Reich/Shafrir [2] (which generalizes Ishikawa’s result to unbounded C), Kohlenbach [17] obtained for the first time explicit bounds on the asymptotic regularity when general sequences (λn) satisfying (3), (4) are considered.”
Effective metastability of Halpern iterates in CAT(0) spaces (2011, methodology): “Note that what is called ‘property I’ and ‘property S’ in [1] has been studied under the name of ‘axioms (W2) and (W4)’ in [21].”