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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 ..."
Abstract

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.
URL:www.emis.de/journals/AFA/ ON THE SUZUKI NONEXPANSIVETYPE MAPPINGS
"... Abstract. It is shown that if C is a nonempty convex and weakly compact subset of a Banach space X with M(X)> 1 and T: C → C satisfies condition (C) or is continuous and satisfies condition (Cλ) for some λ ∈ (0, 1), then T has a fixed point. In particular, our theorem holds for uniformly nonsquare B ..."
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Abstract. It is shown that if C is a nonempty convex and weakly compact subset of a Banach space X with M(X)> 1 and T: C → C satisfies condition (C) or is continuous and satisfies condition (Cλ) for some λ ∈ (0, 1), then T has a fixed point. In particular, our theorem holds for uniformly nonsquare Banach spaces. A similar statement is proved for nearly uniformly noncreasy spaces. 1.