On the computational content of the krasnoselski and ishikawa fixed point theorems

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 2064, Issue , 2001, Pages 119-145

  Kohlenbach, Ulrich ^a  

^a AARHUS UNIVERSITY   (Denmark)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTABILITY AND DECIDABILITY; FUNCTIONAL ANALYSIS; SET THEORY; TOPOLOGY;

APPROXIMATE FIXED POINT; ASYMPTOTIC REGULARITIES; COMPACTNESS CONDITION; COMPUTATIONAL CONTENTS; FIXED POINT THEOREMS; FIXED POINT THEORY; NONLINEAR FUNCTIONAL ANALYSIS; RATE OF CONVERGENCE;

FIXED POINT ARITHMETIC;

EID: 84974728199     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-45335-0_9     Document Type: Conference Paper

References (36)

1. Baillon, J, Bruck, R.E., The rate of asymptotic regularity is 0(1 pn). Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math. 178, pp. 51-81, Dekker, New York, 1996.
2. Bruck, R.E., A simple proof that the rate of asymptotic regularity of (I + T)=2 is O(1=pn). Recent advances on metric fixed point theory (Seville, 1995), pp. 11-18, Ciencias, 48, Univ. Sevilla, Seville, 1996.
3. Bonsall, F.F., Lectures on some fixed point theorems of functional analysis. Tata Institute of Fundamental Research. Bombay 1962.
4. Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992).
5. Browder, F.E., Nonexpansive nonlinear operators in a Banach space. Proc. Nat. Acad. Sci. U.S.A. 54, pp. 1041-1044 (1965).
6. Clarkson, J.A., Uniformly convex spaces. Trans. Amer. Math. Soc. 40, pp. 396-414 (1936).
7. Deimling, K., Nonlinear Functional Analysis. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, xiv+450 pp., 1985.
8. Edelstein, M., O’Brian, R.C., Nonexpansive mappings, asymptotic regularity and successive approximations. J. London Math. Soc. 17, pp. 547-554 (1978).
9. Espinola, R., Kirk, W.A., Fixed points and approximated fixed points in product spaces. Preprint.
10. Feferman, S., Kreisel’s ‘Unwinding Program’. In: P. Odifreddi (ed.), Kreiseliana:about and around Georg Kreisel, A.K. Peters, Wellesley Massachusetts, pp. 247-273 (1996).
11. Goebel, K., Kirk, W.A., Iteration processes for nonexpansive mappings. In: Singh, S.P., Thomeier, S., Watson, B., eds., Topological Methods in Nonlinear Functional Analysis. Contemporary Mathematics 21, AMS, pp. 115-123 (1983).
12. Goebel, K., Kirk, W.A., Topics in metric fixed point theory. Cambridge studies in advanced mathematics 28, Cambridge University Press 1990.
13. Göhde, D., Zum Prinzip der kontraktiven Abbildung. Math. Nachrichten 30, pp. 251-258 (1965).
14. Hanner, O., On the uniform convexity of Lp and lp. Ark. Mat. 3, pp. 239-244 (1956).
15. Henry, M.S., Schmidt, D., Continuity theorems for the product approximation operator. In: Law, A.G., Sahney, B.N. (eds.), Theory of Approximation with Applications, pp. 24-42, Academic Press, New York (1976).
16. Ishikawa, S., Fixed points and iterations of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59, pp. 65-71 (1976).
17. Kirk, W.A., A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72, pp. 1004-1006 (1965).
18. Kirk, W.A., Martinez-Yanez, C., Approximate fixed points for nonexpansive mappings in uniformly convex spaces. Annales Polonici Mathematici 51, pp. 189-193 (1990).
19. Kirk,W.A., Nonexpansive mappings and asymptotic regularity. To appear in: Nonlinear Analysis.
20. Koethe, G., Topoplogische Lineare Räume. Springer-Verlag Berlin-Göttingen-Heidelberg, VI+456 pp., 1960.
21. Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebychefi approximation. Ann. Pure Appl. Logic 64, pp. 27-94 (1993).
22. Kohlenbach, U., 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. Numer. Funct. Anal. and Optimiz. 14, pp. 581-606 (1993).
23. Kohlenbach, U., Analysing proofs in analysis. In: W. Hodges, M. Hyland, C. Steinhorn, J. Truss, editors, Logic: from Foundations to Applications. European Logic Colloquium (Keele, 1993), pp. 225-260, Oxford University Press (1996).
24. Kohlenbach, U., Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals. Arch. Math. Logic 36, pp. 31-71 (1996).
25. Kohlenbach, U., Proof theory and computational analysis. Electronic Notes in Theoretical Computer Science 13, Elsevier (http://www.elsevier.nl/locate/entcs/volume13.html), 34 pages (1998).
26. Kohlenbach, U., Elimination of Skolem functions for monotone formulas in analysis. Arch. Math. Logic 37, pp. 363-390 (1998).
27. Kohlenbach, U., Arithmetizing proofs in analysis. In: Larrazabal, J.M. et al. (eds.), Proceedings Logic Colloquium 96 (San Sebastian), Springer Lecture Notes in Logic 12, pp. 115-158 (1998).
28. Kohlenbach, U., Things that can and things that can’t be done in PRA. Ann. Pure Appl. Logic 102, pp. 223-245 (2000).
29. Kohlenbach, U., A quantitative version of a theorem due to Borwein-Reich-Shafrir. Submitted.
30. Kohlenbach, U., Oliva, P., Effective bounds on strong unicity in L1-approximation. Preprint 30pp. (2001).
31. Krasnoselski, M. A., Two remarks on the method of successive approximation. Usp. Math. Nauk (N.S.) 10, pp. 123-127 (1955) (Russian).
32. Linz, Peter, A critique of numerical analysis. Bull. Amer. Math. Soc. 19, pp. 407-416 (1988).
33. Luckhardt, H., Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken. J. Symbolic Logic 54, pp. 234-263 (1989).
34. Mann, W.R., Mean value methods in iteration. Proc. Amer. Math. Soc. 4, pp. 506-510 (1953).
35. Reich, S., Shafrir, I., Nonexpansive iterations in hyperbolic spaces. Nonlinear Analysis, Theory, Methods and Applications 15, pp. 537-558 (1990).
36. Schwichtenberg, H., Refined Program Extraction from Classical Proofs. To appear in: Proceedings of 1999 Marktobersdorf Summer School.