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Mathematische Zeitschrift

, Volume 271, Issue 3–4, pp 1271–1285 | Cite as

On the approximate fixed point property in abstract spaces

  • C. S. Barroso
  • O. F. K. Kalenda
  • P.-K. Lin
Article

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 1-theorem for 1-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.

Keywords

Weak approximate fixed point property Metrizable locally convex space 1 sequence Fréchet–Urysohn space 

Mathematics Subject Classification (2000)

Primary 47H10 46A03 

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References

  1. 1.
    Archangel’skii, A.V.: Topological Spaces of Functions, Moscow State Univ. (1989) (in Russian). Kluwer Academic Publication, Dordrecht, Boston, London (1992) (in English)Google Scholar
  2. 2.
    Asplund E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barroso C.S.: The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete Cont. Dyn. Syst. 25, 467–479 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barroso C.S., Lin P.-K.: On the weak approximate fixed point property. J. Math. Anal. Appl. 365, 171–175 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 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)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Domínguez Benavides T.: A renorming of some nonseparable Banach spaces with the fixed point property. J. Math. Anal. Appl. 350, 525–530 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bourgain J., Fremlin D.H., Talagrand M.: Pointwise compact sets of Baire measurable functions. Am. J. Math. 100, 845–886 (1978)MathSciNetMATHCrossRefGoogle Scholar
  8. 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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Díaz J.C.: Montel subspaces in the countable projective limits of L p(μ)-spaces. Can. Math. Bull. 32, 169–176 (1989)MATHCrossRefGoogle Scholar
  10. 10.
    Fabian M.: Gâteaux differentiability of convex functions and topology: weak Asplund spaces. Wiley-Interscience, New York (1997)MATHGoogle Scholar
  11. 11.
    Fan K.: Sur un théorème minimax. C. R. Acad. Sci. Paris 259, 3925–3928 (1964)MathSciNetMATHGoogle Scholar
  12. 12.
    Hazewinkel M., van de Vel M.: On almost-fixed point theory. Can. J. Math. 30, 673–699 (1978)MATHCrossRefGoogle Scholar
  13. 13.
    Idzik A.: On γ-almost fixed point theorems. The single-valued case. Bull. Polish. Acad. Sci. Math. 35, 461–464 (1987)MathSciNetMATHGoogle Scholar
  14. 14.
    Idzik A.: Almost fixed point theorems. Proc. Am. Math. Soc. 104, 779–784 (1988)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jafari F., Sehgal V.M.: Some fixed point theorems for nonconvex spaces. Int. J. Math. & Math. Sci. 21, 133–138 (1998)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Johnson W.B., Lindenstrauss J.: Some remarks on weakly compactly generated Banach spaces. Israel J. Math. 17, 219–230 (1974)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kalenda O.F.K.: Valdivia compact spaces in topology and Banach space theory. Extracta Math. 15(1), 1–85 (2000)MathSciNetMATHGoogle Scholar
  18. 18.
    Kalenda O.F.K.: Spaces not containing 1 have weak approximate fixed point property. J. Math. Anal. Appl. 373, 134–137 (2011)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Köthe G.: Topological Vector Spaces I. Springer-Verlag, New York (1969)MATHCrossRefGoogle Scholar
  20. 20.
    Lin P.-K., Sternfeld Y.: Convex sets with the Lipschitz fixed point property are compact. Proc. Am. Math. Soc. 93, 633–639 (1985)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Moloney J., Weng X.: A fixed point theorem for demicontinuous pseudocontractions in Hilbert spaces. Studia Math. 116(3), 217–223 (1995)MathSciNetMATHGoogle Scholar
  22. 22.
    Namioka I., Phelps R.R.: Banach spacess which are Asplund spaces. Duke Math. J. 42(4), 735–750 (1975)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rosenthal H.P.: A characterization of Banach spaces containing 1. Proc. Nat. Acad. Sci. USA 71(6), 2411–2413 (1974)MATHCrossRefGoogle Scholar
  24. 24.
    Tijs S.H., Torre A., Branzei R.: Approximate fixed point theorems. Libertas Math. 23, 35–39 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    Troyanski S.L.: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces. Studia Math. 37, 173–180 (1971)MathSciNetMATHGoogle Scholar
  26. 26.
    van de Vel, M., The intersection property: a contribution to almost fixed point theory, Thesis, University of Imtelling Antewerpen, 1975Google Scholar
  27. 27.
    van der Walt, T.: Fixed and almost fixed points, Thesis, Math. Centre, Amsterdam, 1963Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil
  2. 2.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic
  3. 3.Department of MathematicsUniversity of MemphisMemphisUSA

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