Krasnoselski-Mann Iterations in Normed Spaces

Borwein, Jonathan M.; Reich, Simeon; Shafrir, Itaï

doi:10.4153/cmb-1992-003-0

Cited by 97 publications

(109 citation statements)

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“…Such a Φ has been extracted in [71] (for the normed case already in [62]) from the original ineffective proof in [9]: For various applications of this result (also new qualitative results which only use the uniformity of the bound but not its numerical value) see [71] and [72]. For uniformly convex hyperbolic spaces often special arguments yielding better bounds apply.…”

Section: Theorem 511 ([28])mentioning

confidence: 99%

Gödel's Functional Interpretation and Its Use in Current Mathematics

Kohlenbach

2011

Kurt Gödel and the Foundations of Mathematics

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Section: Theorem 511 ([28])mentioning

confidence: 99%

Gödel's Functional Interpretation and Its Use in Current Mathematics

Kohlenbach

2011

Kurt Gödel and the Foundations of Mathematics

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“…If we add the exponential function to PBA we obtain a system EBA which guarantees that Φ uses at most a finite iteration of exp (so if exp is not iterated at all the bound will be exponential in n relative to x). Whereas for our first application in the present paper (theorem 3.9 below) this result for PBA is already sufficient for providing the general logical framework, our analysis of a proof from [4] carried out in theorem 4.5 needs an extended version due to the use of a principle used in that proof which is not available in PBA or EBA. Whereas these systems contain quite some parts of nonconstructive analysis, principles based on sequential compactness are not included.…”

Section: General Introductionmentioning

confidence: 98%

“…has been proved in certain formal systems T for (fragments of) analysis, then one can extract a computable uniform bound Φ(n, x) 4 for ∃m, i.e.…”

Section: General Introductionmentioning

confidence: 99%

“…of convergence for bounded monotone sequences (a n ) n∈IN of reals, as it is this principle which is used in the proof from [4] we are going to analyse. In systems like PBA, real numbers are represented as Cauchy sequences of rational numbers with fixed rate of convergence.…”

Section: General Introductionmentioning

confidence: 99%

“…We consider strong generalizations of Krasnoselski's result due to [16], [8], [11] and [4]. In [16] it is shown that Krasnoselski's fixed point theorem even holds without the assumption of X being uniformly convex.…”

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confidence: 97%

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Effective Uniform Bounds on the Krasnoselski-Mann Iteration

Kohlenbach¹

2000

BRICS

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This paper is a case study in proof mining applied to non-effective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex spaces the rate of convergence is in general not computable in f (which is related to the non-uniqueness of fixed points). However, the iterations yield approximate fixed points of arbitrary quality for general normed spaces and bounded C (asymptotic regularity). In this paper we apply general proof theoretic results obtained in previous papers to non-effective proofs of this regularity and extract uniform explicit bounds on the rate of the asymptotic regularity. We start off with the classical case of uniformly convex spaces treated already by Krasnoselski and show how a logically motivated modification allows to obtain an improved bound. Already the analysis of the original proof (from 1955) yields an elementary proof for a result which was obtained only in 1990 with the use of the deep Browder-G¨ohde-Kirk fixed point theorem. The improved bound from the modified proof gives applied to various special spaces results which previously had been obtained only by ad hoc calculations and which in some case are known to be optimal. The main section of the paper deals with the general case of arbitrary normed spaces and yields new results including a quantitative analysis of a theorem due to Borwein, Reich and Shafrir (1992) on the asymptotic behaviour of the general Krasnoselski-Mann iteration in arbitrary normed spaces even for unbounded sets C. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function and the normed space X and, in fact, only depend on the error epsilon, an upper bound on the diameter of C and some very general information on the sequence of scalars k used in the iteration. Even non-effectively only the existence of bounds satisfying weaker uniformity conditions was known before except for the special situation, where lambda_k := lambda is constant. For the unbounded case, no quantitative information was known so far.

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