Open Access

Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces

Fixed Point Theory and Applications20062006:59692

https://doi.org/10.1155/FPTA/2006/59692

Received: 19 August 2005

Accepted: 26 February 2006

Published: 1 June 2006

Abstract

We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let be a bounded closed convex subset of a uniformly smooth Banach space . Let be an infinite family of commuting nonexpansive mappings on . Let and be sequences in satisfying for . Fix and define a sequence in by for . Then converges strongly to , where is the unique sunny nonexpansive retraction from onto .

[123456789101112131415161718192021222324252627]

Authors’ Affiliations

(1)
Department of Mathematics, Kyushu Institute of Technology

References

  1. Baillon J-B: Quelques aspects de la théorie des points fixes dans les espaces de Banach. I, II. In Séminaire d'Analyse Fonctionnelle (1978–1979) Exp. No. 7–8. École Polytech, Palaiseau; 1979:45.Google Scholar
  2. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 1922, 3: 133–181.MATHGoogle Scholar
  3. Brodskiĭ MS, Mil'man DP: On the center of a convex set. Doklady Akademii Nauk SSSR (New Series) 1948, 59: 837–840.MathSciNetGoogle Scholar
  4. Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar
  5. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar
  6. Browder FE: Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces. Archive for Rational Mechanics and Analysis 1967,24(1):82–90.MathSciNetView ArticleMATHGoogle Scholar
  7. Bruck RE Jr.: Nonexpansive retracts of Banach spaces. Bulletin of the American Mathematical Society 1970, 76: 384–386. 10.1090/S0002-9904-1970-12486-7MathSciNetView ArticleMATHGoogle Scholar
  8. Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Transactions of the American Mathematical Society 1973, 179: 251–262.MathSciNetView ArticleMATHGoogle Scholar
  9. Bruck RE Jr.: A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific Journal of Mathematics 1974, 53: 59–71.MathSciNetView ArticleMATHGoogle Scholar
  10. Göhde D: Zum Prinzip der kontraktiven Abbildung. Mathematische Nachrichten 1965, 30: 251–258. 10.1002/mana.19650300312MathSciNetView ArticleMATHGoogle Scholar
  11. Gossez J-P, Lami Dozo E: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pacific Journal of Mathematics 1972, 40: 565–573.MathSciNetView ArticleMATHGoogle Scholar
  12. Kelley JL: General Topology. Van Nostrand Reinhold, New York; 1955:xiv+298.Google Scholar
  13. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar
  14. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the American Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar
  15. Reich S: Asymptotic behavior of contractions in Banach spaces. Journal of Mathematical Analysis and Applications 1973,44(1):57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleMATHGoogle Scholar
  16. Reich S: Product formulas, nonlinear semigroups, and accretive operators. Journal of Functional Analysis 1980,36(2):147–168. 10.1016/0022-1236(80)90097-XMathSciNetView ArticleMATHGoogle Scholar
  17. Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar
  18. Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces. Nonlinear Analysis 1998,34(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar
  19. Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proceedings of the American Mathematical Society 2003,131(7):2133–2136. 10.1090/S0002-9939-02-06844-2MathSciNetView ArticleMATHGoogle Scholar
  20. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar
  21. Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory and Applications 2005,2005(1):103–123. 10.1155/FPTA.2005.103View ArticleMathSciNetMATHGoogle Scholar
  22. Suzuki T: The set of common fixed points of an n-parameter continuous semigroup of mappings. Nonlinear Analysis 2005,63(8):1180–1190. 10.1016/j.na.2005.05.035MathSciNetView ArticleMATHGoogle Scholar
  23. Suzuki T: The set of common fixed points of a one-parameter continuous semigroup of mappings is . Proceedings of the American Mathematical Society 2006,134(3):673–681. 10.1090/S0002-9939-05-08361-9MathSciNetView ArticleMATHGoogle Scholar
  24. Suzuki T: Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces. to appear in Israel Journal of MathematicsGoogle Scholar
  25. Suzuki T: Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Hilbert spaces. to appear in Proceedings of the Fourth International Conference on Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka, eds.), Yokohama Publishers, YokohamaGoogle Scholar
  26. Takahashi W, Ueda Y: On Reich's strong convergence theorems for resolvents of accretive operators. Journal of Mathematical Analysis and Applications 1984,104(2):546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar
  27. Turett B: A dual view of a theorem of Baillon. In Nonlinear Analysis and Applications (St. Johns, Nfld., 1981), Lecture Notes in Pure and Appl. Math.. Volume 80. Dekker, New York; 1982:279–286.Google Scholar

Copyright

© Tomonari Suzuki. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.