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IMADA - Department of Mathematics and Computer Science |
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The first part of the talk indicates how techniques from Mathematical Logic, so-called proof interpretations, can be used to extract new information from ineffective proofs in various areas of mathematics and, in particular, in functional analysis. In the second part we present results (partly together with Laurentiu Leustean) of a recent case study where this approach has been applied to proofs in metric fixed point theory. This concerns the asymptotic regularity of various iteration schemes of nonexpansive functions. The results which extend to the general setting of convex metric spaces (Takahashi) resp. hyperbolic spaces (Goebel,Kirk,Reich) and to directionally nonexpansive functions (Kirk), not only provide new effective bounds but even yield systematically new qualitative results on the uniformity of asymptotic regularity. The latter generalize all known results of this kind which had been obtained by functional analytic embedding techniques during the last 20 years. We conclude the talk by presenting a new general logical meta-theorem which implies such uniformity results "a priori" if certain easy to check logical conditions are met. Only to get explicit effective bounds one has to carry out an actual proof interpretation. Host: Klaus Meer
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