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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 (20 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.
Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces
 Fixed Point Theory and Applications
, 2005
"... In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1. ..."
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Cited by 8 (2 self)
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In 1979, Ishikawa proved a strong convergence theorem for finite families of nonexpansive mappings in general Banach spaces. Motivated by Ishikawa’s result, we prove strong convergence theorems for infinite families of nonexpansive mappings. 1.
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 nonsquar ..."
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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.
Convergence Theorems of ThreeStep Iterative Scheme for a Finite Family of Uniformly QuasiLipschitzian Mappings in Convex Metric Spaces
, 2009
"... We consider a new Noortype iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasiLipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite ..."
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We consider a new Noortype iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasiLipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasiLipschitzian mappings. The results presented in this paper extend, improve and unify some main results in previous work. Copyright q 2009 T. Youxian and Y. Chunde. 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.