Results 1 
9 of
9
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 45 (26 self)
 Add to MetaCart
(Show Context)
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.
A quadratic rate of asymptotic regularity for CAT(0)spaces
, 2005
"... In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hy ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch’s theorem obtained by Kohlenbach using methods from mathematical logic (socalled “proof mining”).
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
"... ..."
(Show Context)
Some Remarks On The Convergence Of Picard
, 2006
"... Some examples are given to illustarte that the characterization in [8] for the convergence of Picard iteration to a fixed point for a continuous mapping, is false. Throughout this paper, let T be a mapping from a nonempty subset D of a metric space X into X and let F (T) be a fixed point set of T. A ..."
Abstract
 Add to MetaCart
Some examples are given to illustarte that the characterization in [8] for the convergence of Picard iteration to a fixed point for a continuous mapping, is false. Throughout this paper, let T be a mapping from a nonempty subset D of a metric space X into X and let F (T) be a fixed point set of T. A mapping T: D → X is called nonexpansive if for each x, y ∈ D, d(T (x), T (y)) ≤ d(x, y). T is said to be quasinonexpansive if for each x ∈ D and for every p ∈ F (T), d(T (x), p) ≤ d(x, p). T is conditionally quasinonexpansive if it is quasinonexpansive whenever F (T) 6 = ∅. Since convergence theorems of iterations to a fixed point for nonexpansive mapping are discussed in [12], many results on the convergence of some iterations to fixed points for nonexpansive, quasinonexpansive and generalized types of quasinonexpansive mappings in metric and Banach spaces have appeared (for example, [38]). Following Ghosh and Debnath [4], if D is a convex subset of a normed space X and T: D → D, Ishikawa introduced the following iteration x0 ∈ D, xn = Tnλ,µ(x0), Tλ,µ = (1 − λ)I + λT [(1 − µ)I + µT], for each n ∈ N (the set of all positive integers), where λ ∈ (0, 1) and µ ∈ [0, 1). When µ = 0, it yields that Tλ,µ = Tλ and the iteration becomes x0 ∈ D, xn = Tnλ (x0), Tλ = (1 − λ)I + λT. This iteration is called Mann iteration. If Tµ = (1 − µ)I + µT, Tλ,µ may be written in the form Tλ,µ = (1 − λ)I + λTTµ. Recall a mapping T is asymptotically regular at x0 ∈ D if lim n→ ∞ d(T n(x0), Tn+1(x0)) = 0.
Applied Foundations: Proof Mining in Mathematics
"... A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim ..."
Abstract
 Add to MetaCart
A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim was to show that the use of such ideal elements can be shown to be consistent by finitistic means (‘Hilbert’s program’). Hilbert’s program turned out to be impossible in the original form by the seminal results of K. Gödel. However, more recent developments show it can be carried out in a partial form in that one can design formal systems A which are sufficient to formalize substantial parts of mathematics and yet can be reduced prooftheoretically to primitive recursive arithmetic PRA, a formal system usually associated with ‘finitism’. These systems