Banas–Hajnosz–Wedrychowicz type modulus of convexity and normal structure in Banach spaces



In this paper, we present some sufficient conditions for which the Banach space X has uniform normal structure in terms of the Banas–Hajnosz–Wedrychowicz type modulus of convexity \(SY_X(\epsilon )\), the coefficient of weak orthogonality \(\omega (X)\) and the Domínguez–Benavides coefficient R(1, X). Some known results are improved and strengthened.


Banas–Hajnosz–Wedrychowicz type modulus of convexity coefficient of weak orthogonality Domínguez–Benavides coefficient normal structure 

Mathematics Subject Classification

Primary 46B20 Secondary 47H09 



This research was partially supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1601006), the Chongqing New-star Plan of Science and Technology (KJXX2017012), the Scientific Technological Research Program of the Chongqing Three Gorges University (no. 16PY11), the Chongqing Municipal Key Laboratory of Institutions of Higher Education (Grant no. [2017]3), and the Program of Chongqing Development and Reform Commission (Grant no. 2017[1007]), the Key Laboratory for Nonlinear Science and System Structure, Chongqing Three Georges University.


  1. 1.
    Banas, J., Hajnosz, A., Wedrychowicz, S.: On convexity and smoothness of Banach space. Comment. Math. Univ. Carol. 3, 445–452 (1990)MathSciNetMATHGoogle Scholar
  2. 2.
    Clarkson, J.: Uniform convex spaces. Trans. Am. Math. Soc. 40, 396–414 (1936)CrossRefMATHGoogle Scholar
  3. 3.
    Domínguez Benavides, T.: A geometrical coefficient implying the fixed point property and stability results, Houst. J. Math. 22, 835–849 (1996)MathSciNetMATHGoogle Scholar
  4. 4.
    Gao, J.: Normal structure and modulus of \(U\)-convexity in Banach spaces, function spaces, differential operators and nonlinear analysis (Paseky and Jizerou, 1995), Prometheus, Prague, pp. 195-199 (1996)Google Scholar
  5. 5.
    Goebel, K., Kirk, W.: Topics in metric fixed point theory. In: Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1990)Google Scholar
  6. 6.
    Jiménez-Melado, A., Llorens-Fuster, E., Saejung, S.: The von Neumann–Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc. Am. Math. Soc. 2, 355–364 (2006)MathSciNetMATHGoogle Scholar
  7. 7.
    Khamsi, M.: Uniform smoothness implies super-normal structure property. Nonlinear Anal. 19, 1063–1069 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kirk, W.: A fixed point theorem for mappings which do not increase distances. Am. Math. Monthly 72, 1004–1006 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mazcuñán-Navarro, E.: On the modulus of \(u\)-convexity of Ji Gao. Abstr. Appl. Anal. 1, 49–54 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Prus, S.: Some estimates for the normal structure coefficient in Banach spaces. Rend. Circolo Mat. Palermo 1, 128–135 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Saejung, S.: On the modulus of \(U\)-convexity. Abstr. Appl. Anal. 1, 59–66 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Saejung, S., Gao, J.: On Banas–Hajnosz–Wedrychowicz type modulus of convexity and fixed point property. Nonlinear Funct. Anal. Appl. 4, 717–725 (2016)MATHGoogle Scholar
  13. 13.
    Sims, B.: Ultra-techniques in Banach space theory. In: Queen’s Papers in Pure and Applied Mathematics, vol. 60. Queens University, Kingston (1982)Google Scholar
  14. 14.
    Sims, B.: A class of spaces with weak normal structure. Bull. Aust. Math. Soc. 50, 523–528 (1994)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zuo, Z., Cui, Y.: Some modulus and normal structure in Banach space. J. Inequal. Appl. 1, 1–15 (2009)MathSciNetMATHGoogle Scholar
  16. 16.
    Zuo, Z., Tang, C.: On James and Jordan von-Neumann type constants and the normal structure in Banach spaces. Topol. Methods Nonlinear Anal. 2, 615–623 (2017)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsChongqing Three Gorges UniversityWanzhouPeople’s Republic of China
  2. 2.Key Laboratory of Intelligent Information Processing and Control, School of Computer Science and EngineeringChongqing Three Gorges UniversityWanzhouPeople’s Republic of China

Personalised recommendations