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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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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.
DIAMETRICALLY CONTRACTIVE MAPS AND FIXED POINTS
, 2006
"... Contractive maps have nice properties concerning fixed points; a big amount of literature has been devoted to fixed points of nonexpansive maps. The class of shrinking (or strictly contractive) maps is slightly less popular: few specific results on them (not applicable to all nonexpansive maps) appe ..."
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Cited by 1 (0 self)
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Contractive maps have nice properties concerning fixed points; a big amount of literature has been devoted to fixed points of nonexpansive maps. The class of shrinking (or strictly contractive) maps is slightly less popular: few specific results on them (not applicable to all nonexpansive maps) appear in the literature and some interesting problems remain open. As an attempt to fill this gap, a condition half way between shrinking and contractive maps has been studied recently. Here we continue the study of the latter notion, solving some open problems concerning these maps. Copyright © 2006 Marco Baronti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Some Problems Connected with Diametrically Contractive Maps and Fixed Point Theory With Python Programming Language
"... Abstract: In this paper we introduce nice properties of diametrically contractive maps with fixed point. We have given some counter examples to explain this idea and some interesting problems remain open. We have also discussed a concept of fixed point iteration used in Python programming language. ..."
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Abstract: In this paper we introduce nice properties of diametrically contractive maps with fixed point. We have given some counter examples to explain this idea and some interesting problems remain open. We have also discussed a concept of fixed point iteration used in Python programming language.