Some logical metatheorems with applications in functional analysis

Transactions of the American Mathematical Society

Volumn 357, Issue 1, 2005, Pages 89-128

  Kohlenbach, Ulrich ^a  

^a AARHUS UNIVERSITY   (Denmark)

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[No Author keywords available]

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EID: 12744263577     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-04-03515-9     Document Type: Article

References (55)

1. Beeson, M., Foundations of constructive mathematics. Springer Verlag, Berlin, Heidelberg, New York (1985). MR 86k:03055
2. Bellin, G., Ramsey interpreted: a parametric version of Ramsey's theorem. In: Logic and computation (Pittsburgh, PA, 1987), pp. 17-37, Contemp. Math., 106, Amer. Math. Soc., Providence, RI (1990). MR 911:03106
3. Bezem, M.A., Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals. J. Symb. Logic 50 pp. 652-660 (1985). MR 87c:03139
4. Borwein, J., Reich, S., Shafrir, I., Krasnoselski-Mann iterations in normed spaces. Canad. Math. Bull. 35, pp. 21-28 (1992). MR 93e:47072
5. Bridson, M.R., Haefliger, A., Metric spaces of non-positive curvature. Springer Verlag, Berlin (1999). MR 2000k:53038
6. Browder, F.E., Petryshyn, W.V., The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Amer. Math. Soc. 72, pp. 571-575 (1966). MR 32:8155b
7. Bruhat, F., Tits, J., Groupes réductifs sur un corps local. I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math. 41, pp. 5-251 (1972). MR 48:6265
8. Delzell, C., Kreisel's unwinding of Artin's proof-Part I. In: Odifreddi, P., Kreiseliana, 113-246, A K Peters, Wellesley, MA (1996).
9. Edelstein, M., O'Brien, R.C., Nonexpansive mappings, asymptotic regularity and successive approximations. J. London Math. Soc. 17, pp. 547-554 (1978). MR 80b:47074
10. Friedrich, W., Spielquantorinterpretation unstetiger Funktionale der höheren Analysis. Arch. math. Logik 24, pp. 73-99 (1984). MR 86c:03052
11. Friedrich, W., Gödelsche Funktionalinterpretation für eine Erweiterung der klassischen Analysis. Zeitschr. f. math. Logik und Grundl. d. Math. 31, pp. 3-29 (1985). MR 87b:03132
12. 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). MR 85a:47059
13. Goebel, K., Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Monographs and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York, ix+170 pp. (1984). MR 86d:58012
14. Gödel, K., Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280-287 (1958). MR 21:1275
15. Gödel, K., Collected Work, Vol.2, S. Feferman et al. eds. Oxford University Press, New York (1990). MR 91b:01093
16. Groetsch, C.W., A note on segmenting Mann iterates. J. of Math. Anal. Appl. 40, pp. 369-372 (1972). MR 49:5954
17. Henson, C.W., Iovino, J., Ultraproducts in analysis. In: Analysis and Logic. London Mathematical Society LNS 262, pp. 1-112. Cambridge University Press (2002).
18. Howard, W.A., Hereditarily majorizable functionals of finite type. In: Troelstra (ed.), Metamathematical investigation of intuitionistic arithmetic and analysis, pp. 454-461. Springer LNM 344 (1973). MR 57:9493
19. Howard, W.A., Kreisel, G., Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis. J. Symbolic Logic 31, pp. 325-358 (1966). MR 35:27
20. Ishikawa, S., Fixed points and iterations of a nonexpansive mapping in a Banach space. Proc. Amer. Math. Soc. 59, pp. 65-71 (1976). MR 54:1030
21. Itoh, S., Some fixed point theorems in metric spaces. Fund. Math. 102, pp. 109-117 (1979). MR 80h:54058
22. Kirk, W.A., Krasnosel'skii iteration process in hyperbolic spaces, Numer. Funct. Anal. and Optimiz. 4, pp. 371-381 (1982). MR 84e:47067
23. Kirk, W.A., Nonexpansive mappings and asymptotic regularity, Nonlinear Analysis, 40, pp. 323-332 (2000). MR 2001m:47116
24. Kirk, W.A., Geodesic geometry and fixed point theory. Preprint 23pp. (2003).
25. Kirk, W.A., Martinez-Yanez, C., Approximate fixed points for nonexpansive mappings in uniformly convex spaces. Annales Polonici Mathematici 51, pp. 189-193 (1990). MR 92a:47068
26. Kohlenbach, U., Theorie der majorisierbaren und stetigen Funktionale und ihre Anwendung bei der Extraktion von Schranken aus inkonstruktiven Beweisen: Effektive Eindeutigkeitsmodule bei besten Approximationen aus ineffektiven Beweisen. Ph.D. Thesis, Frankfurt am Main, xxii+278pp. (1990).
27. Kohlenbach, U., Pointwise hereditary majorization and some applications. Arch. Math. Logic 31, pp. 227-241 (1992). MR 93b:03106
28. Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation. Ann. Pure Appl. Logic 64, pp. 27-94 (1993). MR 95b:03061
29. 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). MR 94k:41054
30. 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). MR 98b:03085
31. Kohlenbach, U., Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals. Arch. Math. Logic 36, pp. 31-71 (1996). MR 98h:03073
32. Kohlenbach, U., Arithmetizing proofs in analysis. In: Larrazabal, J.M., Lascar, D., Mints, G. (eds.), Logic Colloquium '96, Springer Lecture Notes in Logic 12, pp. 115-158 (1998). MR 2000j;03085
33. Kohlenbach, U., A note on Spector's quantifier-free rule of extensionality. Arch. Math. Logic 40, pp. 89-92 (2001). MR 2001m:03121
34. Kohlenbach, U., On the computational content of the Krasnoselski and Ishikawa fixed point theorems. In: Proceedings of the Fourth Workshop on Computability and Complexity in Analysis, J. Blanck, V. Brattka, P. Hertling (eds.), Springer LNCS 2064, pp. 119-145 (2001). MR 2003b:47090
35. Kohlenbach, U., A quantitative version of a theorem due to Borwein-Reich-Shafrir. Numer. Funct. Anal. and Optimiz. 22, pp. 641-656 (2001). MR 20021:47076
36. Kohlenbach, U., Uniform asymptotic regularity for Mann iterates. J. Math. Anal. Appl. 279, pp. 531-544 (2003).
37. Kohlenbach, U., Leustean, L., Mann iterates of directionally nonexpansive mappings in hyperbolic spaces. Abstract and Applied Analysis, vol. 2003, issue 8, pp. 449-477 (2003).
38. 1-approximation. Ann. Pure Appl. Logic 121, pp. 1-38 (2003).
39. Kohlenbach, U., Oliva, P., Proof mining: a systematic way of analysing proofs in mathematics. Proc. Steklov Inst. Math. 242, pp. 136-164 (2003).
40. Krasnoselski, M. A., Two remarks on the method of successive approximation. Usp. Math. Nauk (N.S.) 10, pp. 123-127 (1955) (Russian). MR 16:833a
41. 2-formulas. Proc. of the Herbrand symposium (Marseille, 1981), North-Holland (Amsterdam), pp. 39-55 (1982). MR 86b:03076
42. Kreisel, G., Proof Theory and Synthesis of Programs: Potentials and Limitations. Eurocal '85 (Linz 1985), Springer LNCS 203, pp. 136-150 (1986). MR 87e:03156
43. Kreisel, G., Macintyre, A., Constructive logic versus algebraization I. Proc. L.E.J. Brouwer Centenary Symposium (Noordwijkerhout 1981), North-Holland (Amsterdam), pp. 217-260 (1982). MR85j:03108
44. Luckhardt, H., Extensional Gödel Functional Interpretation. Springer Lecture Notes in Mathematics 306 (1973). MR 49:2281
45. Luckhardt, H., Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken. J. Symbolic Logic 54, pp. 234-263 (1989). MR 90k:03056
46. Luckhardt, H., Bounds extracted by Kreisel from ineffective proofs. In: Odifreddi, P., Kreiseliana, 289-300, A K Peters, Wellesley, MA (1996).
47. Machado, H.V., A characterization of convex subsets of normed spaces. Kodai Math. Sem. Rep. 25, pp. 307-320 (1973). MR 48:4703
48. Mann, W.R., Mean value methods in iteration. Proc. Amer. Math. Soc. 4, pp. 506-510 (1953). MR 14:988f
49. Reich, S., Shafrir, I., Nonexpansive iterations in hyperbolic spaces. Nonlinear Analysis, Theory, Methods and Applications 16, pp. 537-558 (1990). MR 91k:47135
50. Reich, S., Zaslavski, A.J., Generic aspects of metric fixed point theory In: W. A. Kirk, B. Sims, editors, Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, pp. 557-576 (2001). MR 2003e:54048
51. Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, xiv+445 pp. (1999). MR 2001i:03126
52. Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1-27 (1962). MR 27:4745
53. Takahashi, W., A convexity in metric space and nonexpansive mappings, I. Kodai Math. Sem. Rep. 22, pp. 142-149 (1970). MR 42:2467
54. Troelstra, A.S. (ed.), Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973). MR 48:3699
55. Weidmann, J., Linear operators in Hubert spaces. Springer Graduate Texts in Mathematics 68, xii+402 pp. (1980). MR 81e:47001