Skip to main content
SpringerLink
Log in

On the approximate fixed point property in abstract spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let X be a Hausdorff topological vector space, X * its topological dual and Z a subset of X *. In this paper, we establish some results concerning the σ(X, Z)-approximate fixed point property for bounded, closed convex subsets C of X. Three major situations are studied. First, when Z is separable in the strong topology. Second, when X is a metrizable locally convex space and Z = X *, and third when X is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet–Urysohn property for certain sets with regarding the σ(X, Z)-topology. The support tools include the Brouwer’s fixed point theorem and an analogous version of the classical Rosenthal’s ℓ 1-theorem for ℓ 1-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Archangel’skii, A.V.: Topological Spaces of Functions, Moscow State Univ. (1989) (in Russian). Kluwer Academic Publication, Dordrecht, Boston, London (1992) (in English)

  2. Asplund E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barroso C.S.: The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont. Dyn. Syst. 25, 467–479 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barroso C.S., Lin P.-K.: On the weak approximate fixed point property. J. Math. Anal. Appl. 365, 171–175 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Domínguez Benavides T., Japon Pineda M.A., Prus S.: Weak compactness and fixed point property for affine mappings. J. Funct. Anal. 209(1), 1–15 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Domínguez Benavides T.: A renorming of some nonseparable Banach spaces with the fixed point property. J. Math. Anal. Appl. 350, 525–530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgain J., Fremlin D.H., Talagrand M.: Pointwise compact sets of Baire measurable functions. Am. J. Math. 100, 845–886 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brânzei R., Morgan J., Scalzo V.: Approximate fixed point theorems in Banach spaces with applications in game theory. J. Math. Anal. Appl. 285, 619–628 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Díaz J.C.: Montel subspaces in the countable projective limits of L p (μ)-spaces. Can. Math. Bull. 32, 169–176 (1989)

    Article  MATH  Google Scholar 

  10. Fabian M.: Gâteaux differentiability of convex functions and topology: weak Asplund spaces. Wiley-Interscience, New York (1997)

    MATH  Google Scholar 

  11. Fan K.: Sur un théorème minimax. C. R. Acad. Sci. Paris 259, 3925–3928 (1964)

    MathSciNet  MATH  Google Scholar 

  12. Hazewinkel M., van de Vel M.: On almost-fixed point theory. Can. J. Math. 30, 673–699 (1978)

    Article  MATH  Google Scholar 

  13. Idzik A.: On γ-almost fixed point theorems. The single-valued case. Bull. Polish. Acad. Sci. Math. 35, 461–464 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Idzik A.: Almost fixed point theorems. Proc. Am. Math. Soc. 104, 779–784 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jafari F., Sehgal V.M.: Some fixed point theorems for nonconvex spaces. Int. J. Math. & Math. Sci. 21, 133–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Johnson W.B., Lindenstrauss J.: Some remarks on weakly compactly generated Banach spaces. Israel J. Math. 17, 219–230 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kalenda O.F.K.: Valdivia compact spaces in topology and Banach space theory. Extracta Math. 15(1), 1–85 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Kalenda O.F.K.: Spaces not containing ℓ 1 have weak approximate fixed point property. J. Math. Anal. Appl. 373, 134–137 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Köthe G.: Topological Vector Spaces I. Springer-Verlag, New York (1969)

    Book  MATH  Google Scholar 

  20. Lin P.-K., Sternfeld Y.: Convex sets with the Lipschitz fixed point property are compact. Proc. Am. Math. Soc. 93, 633–639 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moloney J., Weng X.: A fixed point theorem for demicontinuous pseudocontractions in Hilbert spaces. Studia Math. 116(3), 217–223 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Namioka I., Phelps R.R.: Banach spacess which are Asplund spaces. Duke Math. J. 42(4), 735–750 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rosenthal H.P.: A characterization of Banach spaces containing ℓ 1. Proc. Nat. Acad. Sci. USA 71(6), 2411–2413 (1974)

    Article  MATH  Google Scholar 

  24. Tijs S.H., Torre A., Branzei R.: Approximate fixed point theorems. Libertas Math. 23, 35–39 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Troyanski S.L.: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces. Studia Math. 37, 173–180 (1971)

    MathSciNet  MATH  Google Scholar 

  26. van de Vel, M., The intersection property: a contribution to almost fixed point theory, Thesis, University of Imtelling Antewerpen, 1975

  27. van der Walt, T.: Fixed and almost fixed points, Thesis, Math. Centre, Amsterdam, 1963

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Ceará, Bl 914, Campus do Pici, Fortaleza, CE, 60455-760, Brazil

    C. S. Barroso

  2. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    O. F. K. Kalenda

  3. Department of Mathematics, University of Memphis, Memphis, TN, 38152, USA

    P.-K. Lin

Authors
  1. C. S. Barroso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. F. K. Kalenda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P.-K. Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. S. Barroso.

Additional information

C. S. Barroso’s research has been partially supported by the Brazilian-CNPq Grant. O. F. K. Kalenda supported in part by the grant GAAV IAA 100190901 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education. A very preliminary version of this work was presented at the 4th ENAMA (National Meeting on Mathematical Analysis and Applications) in Belém-PA, Brazil.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barroso, C.S., Kalenda, O.F.K. & Lin, PK. On the approximate fixed point property in abstract spaces. Math. Z. 271, 1271–1285 (2012). https://doi.org/10.1007/s00209-011-0915-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0915-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation