## A quantitative version of a theorem due to Borwein-Reich-Shafrir (2000)

Venue: | Numerical Functional Analysis and Optimization |

Citations: | 19 - 13 self |

### BibTeX

@ARTICLE{Kohlenbach00aquantitative,

author = {Ulrich Kohlenbach},

title = {A quantitative version of a theorem due to Borwein-Reich-Shafrir},

journal = {Numerical Functional Analysis and Optimization},

year = {2000},

volume = {22},

pages = {200--1}

}

### OpenURL

### Abstract

We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann iteration for nonexpansive selfmappings of convex sets in arbitrary normed spaces. 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, but only depend on the error #, an upper bound on the diameter of C and some very general information on the sequence of scalars # k used in the iteration. Only in the special situation, where # k := # is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was ...

### Citations

73 |
Mean value methods in iteration
- Mann
- 1953
(Show Context)
Citation Context ...C # X a non-empty convex subset of X and f : C # C a nonexpansive mapping. We consider the so-called Krasnoselski-Mann iteration (which is more general than the Krasnoselski iteration and due to Mann =-=[23]-=-) generated starting from an arbitrary x # C by x 0 := x, x k+1 := (1 - # k )x k + # k f(x k ), where (# k ) k#IN is a sequence of real numbers in [0, 1]. For background information on this iteration ... |

50 |
Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Monographs and Textbooks
- Goebel, Reich
- 1984
(Show Context)
Citation Context ...at quantitative results a logical analysis of that proof would provide. 2) The results of Ishikawa and Borwein-Reich-Shafrir even hold in the more general setting of hyperbolic spaces in the sense of =-=[10]-=- (see e.g. [4],[27]). We expect that the quantitative analysis carried out in the present paper extends to this setting in a suitable form. 3) In [22] we have shown that the rate of convergence of the... |

49 | Analysing proofs in analysis
- Kohlenbach
- 1993
(Show Context)
Citation Context ...roximation theory) even from highly non-constructive uniqueness proofs and how e#ective rates of convergence can be obtained using this information (see [18] for an introduction to this and [15],[16],=-=[17]-=- for 1 For a di#erent case study in analysis in the context of best approximation theory see [15],[16]. For general information on `proof mining' in analysis see [17],[18]. 2 See also [22] for an inte... |

32 | A fixed point theorem for mappings which do not increase distances - Kirk - 1965 |

30 | Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation
- Kohlenbach
- 1993
(Show Context)
Citation Context ...cheff approximation theory) even from highly non-constructive uniqueness proofs and how effective rates of convergence can be obtained using this information (see [20] for an introduction to this and =-=[17]-=-,[18],[19],[23] 1 For other case studies in analysis in the context of best approximation theory see [17],[18],[23]. For general information on ‘proof mining’ in analysis see [19],[20]. 2 See also [25... |

27 |
rasnoselski-Mann iterations in normed spaces
- Borwein, Reich, et al.
- 1992
(Show Context)
Citation Context ... possibility to extract an algorithm for n in (#) uniformly in x 0 and f (if X,C have a computable representation). We consider strong generalizations of Krasnoselski's result due to [11],[5],[7] and =-=[3]-=-. In [11] it is shown that Krasnoselski's fixed point theorem even holds without the assumption of X being uniformly convex. Even much more general so-called Krasnoselski-Mann iterations x k+1 := (1 -... |

26 |
Topics in metric fixed point theory. Cambridge studies in advanced mathematics 28
- Goebel, Kirk
- 1990
(Show Context)
Citation Context ...p.191) case of uniformly convex spaces due to Krasnoselski. For bounds which, moreover, only depend on C via dC (corollary 2.9) not even the ine#ective existence was known so far and in fact still in =-=[8]-=- (p.101) conjectured as `unlikely' to be true (incidentally by the same authors whose proof of #x k - f(x k )# # 0 in [7] does yield such a bound by logical analysis!). Only in the special case of # k... |

25 |
Fixed points and iteration of a nonexpansive mapping in a Banach space
- Ishikawa
- 1976
(Show Context)
Citation Context ...f f and f is continuous it is clear that for a su#ciently large n on, xm (m # n) will be an approximate fixed point: (#)## > 0#n # IN#m # n(#x m - f(xm )#s#). 3 Due to a much more general result from =-=[11]-=-, which we will discuss below, the assumption of X being uniformly convex actually is superfluous. 3 Because of the simple monotonicity property (see lemma 2.4.1) below) #xm+1 - f(xm+1 )# # #xm - f(xm... |

19 |
Iteration processes for nonexpansive mappings
- Goebel, Kirk
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(Show Context)
Citation Context ... for the possibility to extract an algorithm for n in (#) uniformly in x 0 and f (if X,C have a computable representation). We consider strong generalizations of Krasnoselski's result due to [11],[5],=-=[7]-=- and [3]. In [11] it is shown that Krasnoselski's fixed point theorem even holds without the assumption of X being uniformly convex. Even much more general so-called Krasnoselski-Mann iterations x k+1... |

16 | Proof theory and computational analysis
- Kohlenbach
- 1998
(Show Context)
Citation Context ...of strong unicity as used e.g. in Chebyche# approximation theory) even from highly non-constructive uniqueness proofs and how e#ective rates of convergence can be obtained using this information (see =-=[18]-=- for an introduction to this and [15],[16],[17] for 1 For a di#erent case study in analysis in the context of best approximation theory see [15],[16]. For general information on `proof mining' in anal... |

15 | New effective moduli of uniqueness and uniform a–priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory
- Kohlenbach
- 1993
(Show Context)
Citation Context ... approximation theory) even from highly non-constructive uniqueness proofs and how effective rates of convergence can be obtained using this information (see [20] for an introduction to this and [17],=-=[18]-=-,[19],[23] 1 For other case studies in analysis in the context of best approximation theory see [17],[18],[23]. For general information on ‘proof mining’ in analysis see [19],[20]. 2 See also [25] for... |

12 |
Nonexpansive mappings, asymptotic regularity and successive approximations
- Edelstein, O'Brian
- 1978
(Show Context)
Citation Context ...cial for the possibility to extract an algorithm for n in (#) uniformly in x 0 and f (if X,C have a computable representation). We consider strong generalizations of Krasnoselski's result due to [11],=-=[5]-=-,[7] and [3]. In [11] it is shown that Krasnoselski's fixed point theorem even holds without the assumption of X being uniformly convex. Even much more general so-called Krasnoselski-Mann iterations x... |

12 |
A critique of numerical analysis
- Linz
- 1988
(Show Context)
Citation Context ...d [15],[16],[17] for 1 For a di#erent case study in analysis in the context of best approximation theory see [15],[16]. For general information on `proof mining' in analysis see [17],[18]. 2 See also =-=[22]-=- for an interesting discussion of this and related points. 2 concrete applications to approximation theory). In this paper we are concerned with applications to the first of the two steps mentioned ab... |

11 |
Two remarks on the method of successive approximation
- Krasnoselski
- 1955
(Show Context)
Citation Context ...t theorem, see [4],[9],[12]). If X is a uniformly convex Banach space, C # X is closed convex and bounded and f(C) is a compact subset of C, then a well-known fixed point theorem due to Krasnoselski (=-=[21]-=-) states that a fixed point of f can be approximated by the following Krasnoselski iteration 3 x k+1 := 1 2 (x k + f(x k )), x 0 # C arbitrary. However, the situation still is quite di#erent from the ... |

11 |
The rate of asymptotic regularity is 0( 1 √ n ). Theory and applications of nonlinear operators of accretive and monotone type
- Baillon, Bruck
- 1996
(Show Context)
Citation Context ...theorem even holds without the assumption of X being uniformly convex. Moreover, very general so-called KrasnoselskiMann iterations xk+1 := (1 − λk)xk + λkf(xk) are allowed, where λk is a sequence in =-=[0,1]-=- which is divergent in sum and satisfies lim sup λk < 1. k→∞ In particular, it is proved in [13] that for such iterations (I) lim k→∞ ‖xk − f(xk)‖ = 0, where X is an arbitrary normed linear space, C a... |

11 | On the computational content of the Krasnoselski and Ishikawa fixed point theorems
- Kohlenbach
(Show Context)
Citation Context ...ndation. 1found by applying that method to the original proof of the Borwein/Reich/Shafrir theorem. The general logical method which led to these results will be discussed (with further examples) in =-=[22]-=-. 1 Introduction This paper is the offspring of a case study in the project of analyzing non-effective proofs in analysis by logical tools with the aim of extracting new numerically relevant informati... |

10 |
E#ective moduli from ine#ective uniqueness proofs. An unwinding of de La Vallee Poussin's proof for Chebyche# approximation
- Kohlenbach
- 1993
(Show Context)
Citation Context ...byche# approximation theory) even from highly non-constructive uniqueness proofs and how e#ective rates of convergence can be obtained using this information (see [18] for an introduction to this and =-=[15]-=-,[16],[17] for 1 For a di#erent case study in analysis in the context of best approximation theory see [15],[16]. For general information on `proof mining' in analysis see [17],[18]. 2 See also [22] f... |

10 | On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces - Baillon, Bruck, et al. - 1978 |

8 |
Approximate fixed points for nonexpansive mappings in uniformly convex spaces
- Kirk, Martinez-Yanez
- 1990
(Show Context)
Citation Context ... known before (see [7] where a non-trivial functional theoretic embedding is used to obtain this uniformity after #x k - f(x k )# # 0 has been established by the proof we are analysing) 13 . In fact, =-=[13]-=- explicitly mentions the non-e#ectivity of all these results and states that `it seems unlikely that such estimates would be easy to obtain in general setting' (p.191) and therefore only studies the s... |

8 |
New e#ective moduli of uniqueness and uniform a--priori estimates for constants of strong unicity by logical analysis of known proofs in best approximation theory
- Kohlenbach
- 1993
(Show Context)
Citation Context ...# approximation theory) even from highly non-constructive uniqueness proofs and how e#ective rates of convergence can be obtained using this information (see [18] for an introduction to this and [15],=-=[16]-=-,[17] for 1 For a di#erent case study in analysis in the context of best approximation theory see [15],[16]. For general information on `proof mining' in analysis see [17],[18]. 2 See also [22] for an... |

8 |
Nonexpansive iterations in hyperbolic spaces
- Reich, Shafrir
(Show Context)
Citation Context ...# C and the Krasnoselski-Mann iteration (x n ) starting from x we have #x n - f(x n )# n## # r C (f). 6 With the additional assumption that #k is bounded away from zero, this result is also proved in =-=[24]-=-. 6 Corollary 2.6 ([11],[7],[3]) 7 Under the assumptions of theorem 2.5 plus the additional assumption that C has bounded diameter d(C)s# the following holds: #x # C## > 0#nIN#m # n(#xm - f(xm )# # #)... |

6 |
The rate of asymptotic regularity is 0( # n ). Theory and applications of nonlinear operators of accretive and monotone type
- Baillon, Bruck
- 1996
(Show Context)
Citation Context ...en holds without the assumption of X being uniformly convex. Even much more general so-called Krasnoselski-Mann iterations x k+1 := (1 - # k )x k + # k f(x k ) are allowed, where # k is a sequence in =-=[0, 1]-=- which is divergent in sum and satisfies lim sup k## # ks1. In particular, it is proved in [11] that for such iterations (I) lim k## #x k - f(x k )# = 0, where X is an arbitrary normed linear space, C... |

6 |
Nonexpansive mappings and asymptotic regularity
- Kirk
(Show Context)
Citation Context ...complicated method does not extend to the case of non-constant sequences (# k ). Our bound for the general case of unbounded C treated in [3] (theorem 2.7) is apparently all new. Very recently, Kirk (=-=[14]-=-) obtained a new proof of the uniform (w.r.t. x 0 and f) Ishikawa result for the special case # k = # (again using a functional theoretic embedding) even in the more general setting of so-called direc... |

5 | Nonexpansive nonlinear operators in a Banach - Browder - 1965 |

5 |
Zum Prinzip der kontraktiven Abbildung
- Gohde
- 1965
(Show Context)
Citation Context ...ions which are Lipschitz continuous with Lipschitz constant # = 1), in general fixed points only exist if C is closed convex and bounded (by the famous Browder-Gohde-Kirk fixed point theorem, see [4],=-=[9]-=-,[12]). If X is a uniformly convex Banach space, C # X is closed convex and bounded and f(C) is a compact subset of C, then a well-known fixed point theorem due to Krasnoselski ([21]) states that a fi... |

4 |
Continuity theorems for the product approximation operator
- Henry, Schmidt
- 1976
(Show Context)
Citation Context ...stant of unicity for compact sets K of functions f # C[a, b], if inf f#K dist(f, H) > 0 (H a Haar space), a fact that was proved in approximation theory only in 1976 without providing any bounds (see =-=[10]-=-). 5 and t # (0, 1]. f t : C # C is a contraction and therefore Banach's fixed point theorem applies. Note furthermore that the completeness assumption in Banach's theorem is needed only to guarantee ... |

3 |
A simple proof that the rate of asymptotic regularity of (I + T )/2 is O(1/ # n). Recent advances on metric fixed point theory (Seville
- Bruck
- 1995
(Show Context)
Citation Context ...ee [1], where again the noneffectivity of all known proofs of the full Ishikawa result is stressed) and only for λk := 1 2 a classically proved result of that type has been obtained subsequently (see =-=[3]-=-). This result, of course, is numerically better than our exponential bound in corollary 2.11 when specialised to λ = 1 2 . However, as the authors concede, their extremely complicated method does not... |

3 |
Fixed points and approximate fixed points in product spaces
- Espínola, Kirk
(Show Context)
Citation Context ...e quantitative analysis carried out in the present paper extends to this setting in a suitable form. 3) In [22] we have shown that the rate of convergence of the Krasnoselski iteration towards 14 See =-=[7]-=- for a recent interesting application of this uniformity. 13a fixed point (in the compact case) cannot be computed uniformly in the nonexpansive function f. This phenomenon, which already appears in ... |

2 | Effective bounds on strong unicity - Kohlenbach, Oliva - 2001 |

2 |
Convergence of Krasnoselskii-Mann iterations of nonexpansive operators
- Reich, Zaslavski
- 2000
(Show Context)
Citation Context ...pute unique solutions of functional equations in rather general settings by extracting certain effective data (‘strong uniqueness’) from the uniqueness proof. In this connection the recent results in =-=[28]-=- on cases where the existence of a unique fixed point of certain nonexpansive operators is guaranteed are of interest to apply this logical methodology to in order to possibly get computable bounds on... |

1 |
On the computational content of the Krasnoselski fixed point theorem
- Kohlenbach
(Show Context)
Citation Context ... present the main result of our case study in ordinary mathematical terms without any reference to the general logical mechanism which produced it. For the latter we refer to another publication (see =-=[20]-=-) where it is shown that the result is just an instance of a quite universal scheme. The proof we are going to treat in this paper is taken from the fixed point theory of nonexpansive mappings f : C #... |

1 |
Nonexpansive mappings and asymptotic regularity. Lakshmikantham’s legacy: a tribute on his 75th birthday
- Kirk
- 2000
(Show Context)
Citation Context ... the case of non-constant sequences (λk). Our result in theorem 2.7 on the general case of unbounded C (as treated in [4]) is apparently all new. 4 Final comments and open problems 1) Recently, Kirk (=-=[15]-=-) obtained a new proof of the uniform (w.r.t. x0 and f) Ishikawa result for the special case λk = λ (again using a functional theoretic embedding) even in the more general setting of so-called directi... |