Abstract and Applied Analysis

Modulus of Convexity, the Coeffcient R ( 1 , X ) , and Normal Structure in Banach Spaces

Hongwei Jiao, Yunrui Guo, and Fenghui Wang

Full-text: Open access

Abstract

Let δ X ( ϵ ) and R ( 1 , X ) be the modulus of convexity and the Domínguez-Benavides coefficient, respectively. According to these two geometric parameters, we obtain a sufficient condition for normal structure, that is, a Banach space X has normal structure if 2 δ X ( 1 + ϵ ) > max { ( R ( 1 , x ) - 1 ) ϵ , 1 - ( 1 - ϵ / R ( 1 , X ) - 1 ) } for some ϵ [ 0 , 1 ] which generalizes the known result by Gao and Prus.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 135873, 5 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969174

Digital Object Identifier
doi:10.1155/2008/135873

Mathematical Reviews number (MathSciNet)
MR2411041

Zentralblatt MATH identifier
1166.46302

Citation

Jiao, Hongwei; Guo, Yunrui; Wang, Fenghui. Modulus of Convexity, the Coeffcient $R(1,X)$ , and Normal Structure in Banach Spaces. Abstr. Appl. Anal. 2008 (2008), Article ID 135873, 5 pages. doi:10.1155/2008/135873. https://projecteuclid.org/euclid.aaa/1220969174


Export citation

References

  • S. Dhompongsa, A. Kaewkhao, and S. Tasena, ``On a generalized James constant,'' Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 419--435, 2003.
  • J. Gao, ``Modulus of convexity in Banach spaces,'' Applied Mathematics Letters, vol. 16, no. 3, pp. 273--278, 2003.
  • A. Jiménez-Melado, E. Llorens-Fuster, and S. Saejung, ``The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces,'' Proceedings of the American Mathematical Society, vol. 134, no. 2, pp. 355--364, 2006.
  • E. M. Mazcuñán-Navarro, ``Banach space properties sufficient for normal structure,'' Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 197--218, 2008.
  • S. Saejung, ``The characteristic of convexity of a Banach space and normal structure,'' Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 123--129, 2008.
  • K. Goebel, ``Convexivity of balls and fixed-point theorems for mappings with nonexpansive square,'' Compositio Mathematica, vol. 22, pp. 269--274, 1970.
  • J. A. Clarkson, ``Uniformly convex spaces,'' Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396--414, 1936.
  • S. Prus, ``Some estimates for the normal structure coefficient in Banach spaces,'' Rendiconti del Circolo Matematico di Palermo, vol. 40, no. 1, pp. 128--135, 1991.
  • T. Domínguez Benavides, ``A geometrical coefficient implying the fixed point property and stability results,'' Houston Journal of Mathematics, vol. 22, no. 4, pp. 835--849, 1996.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
  • M. Kato, L. Maligranda, and Y. Takahashi, ``On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces,'' Studia Mathematica, vol. 144, no. 3, pp. 275--295, 2001.