Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

U-FLATNESS AND NON-EXPANSIVE MAPPINGS IN BANACH SPACES

  • Gao, Ji (Department of Mathematics Community College of Philadelphia) ;
  • Saejung, Satit (Department of Mathematics Faculty of Science Khon Kaen University)
  • Received : 2016.01.30
  • Published : 2017.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we define the modulus of n-dimensional U-flatness as the determinant of an $(n+1){\times}(n+1)$ matrix. The properties of the modulus are investigated and the relationships between this modulus and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for non-expansive mappings and normal structure in Banach spaces are obtained.

Keywords

References

  1. B. Bollobas, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181-182. https://doi.org/10.1112/blms/2.2.181
  2. M. S. Brodskii and D. P. Mil'man, On the center of a convex set, (Russian) Doklady Akad. Nauk SSSR (N.S.) 59 (1948), 837-840.
  3. D. Dacunha-Castelle and J. L. Krivine, Applications des ultraproduits a l'etude des espaces et des algebres de Banach, (French) Studia Math. 41 (1972), 315-334. https://doi.org/10.4064/sm-41-3-315-334
  4. M. M. Day, Normed Linear Spaces, Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. Springer-Verlag, New York-Heidelberg, 1973.
  5. J. Diestel, Geometry of Banach spacesselected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975.
  6. J. Gao and S. Saejung, The n-dimensional U-convexity and geometry of Banach spaces, J. Fixed Point Theory 16 (2015), no. 2, 381-392.
  7. R. C. James, Weakly compact sets, Trans. Amer. Math. Soc. 113 (1964), 129-140. https://doi.org/10.1090/S0002-9947-1964-0165344-2
  8. M. A. Khamsi, Uniform smoothness implies super-normal structure property, Nonlinear Anal. 19 (1992), no. 11, 1063-1069. https://doi.org/10.1016/0362-546X(92)90124-W
  9. W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345
  10. T.-C. Lim, On moduli of k-convexity, Abstr. Appl. Anal. 4 (1999), no. 4, 243-247. https://doi.org/10.1155/S1085337599000202
  11. S. Saejung, Convexity conditions and normal structure of Banach spaces, J. Math. Anal. Appl. 344 (2008), no. 2, 851-856. https://doi.org/10.1016/j.jmaa.2008.03.036
  12. B. Sims, "Ultra"-techniques in Banach space theory, Queen's Papers in Pure and Applied Mathematics, Vol. 60, Queen's University, Kingston, ON, 1982.