Nonlinear Functional Analysis and Applications

  • 제26권2호
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  • Pages.433-442
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  • 2021
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
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NORMAL STRUCTURE, FIXED POINTS AND MODULUS OF n-DIMENSIONAL U-CONVEXITY IN BANACH SPACES X AND X*

  • Gao, Ji (Department of Mathematics, Community College of Philadelphia)
  • 투고 : 2020.11.16
  • 심사 : 2021.02.11
  • 발행 : 2021.06.15
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초록

Let X and X* be a Banach space and its dual, respectively, and let B(X) and S(X) be the unit ball and unit sphere of X, respectively. In this paper, we study the relation between Modulus of n-dimensional U-convexity in X* and normal structure in X. Some new results about fixed points of nonexpansive mapping are obtained, and some existing results are improved. Among other results, we proved: if X is a Banach space with $U^n_{X^*}(n+1)>1-{\frac{1}{n+1}}$ where n ∈ ℕ, then X has weak normal structure.

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과제정보

The author would like to thank referees for some suggestions.

참고문헌

  1. A. Aksoy and M.A. Khamsi, Nonstandard methods in fixed point theory, Universitext. New York etc.: Springer-Verlag, 1990.
  2. 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
  3. 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.
  4. J.A. Clarkson, Unifom convex spaces, Trans. Amer. Math. Soc., 40(3) (1936), 396-414. https://doi.org/10.1090/S0002-9947-1936-1501880-4
  5. M.M. Day, Normed linear spaces. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. Springer-Verlag, New York-Heidelberg, 1973.
  6. S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl., 285(2) (2003), 419-435. https://doi.org/10.1016/S0022-247X(03)00408-6
  7. J. Diestel, The Geometry of Banach Spaces - Selected Topics Lecture Notes in Math., 485, Springer - Verlag, Berlin and New York, 1975.
  8. J. Diestel, Sequeces and series in a Banach space, Graduate Texts in Mathematics, 92. New York-Heidelberg-Berlin: Springer-Verlag, 1984.
  9. J. Gao, Normal structure and modulus of U-convexity in Banach spaces. Function spaces, differential operators and nonlinear analysis (Paseky nad Jizerou, 1995), 195-199, Prometheus, Prague, 1996.
  10. J. Gao, Modulus of 2-dimensional U-Convexity and the Geometry of Banach Spaces, J. Nonlinear and Convex Anal., 20(10) (2019), 2041-2051.
  11. J. Gao and S. Saejung, A constant related to fixed points and normal structure in Banach spaces, Nonlinear Funct. Anal. Appl., 16(1) (2011), 17-28.
  12. V.I. Gurariii, Differential properties of convexity moduli of Banach spaces, Mat. Issled., 2(1) (1967), 141-148. (Russian). MR 35*2127. Zbl 232.46024.
  13. R.C. James, Weakly compact sets, Trans. Amer. Math. Soc., 113 (1964), 129-140. https://doi.org/10.1090/S0002-9947-1964-0165344-2
  14. M.A. Khamsi, Uniform smoothness implies super-normal structure property. Nonlinear Anal., 19(1) (1992), 1063-1069. https://doi.org/10.1016/0362-546X(92)90124-W
  15. M.A. Khamsi and B. Sims, Ultra-methods in metric fixed point theory. Kirk, William A. (ed.) et al., Handbook of metric fixed point theory. Dordrecht: Kluwer Academic Publishers. (2001), 177-199.
  16. 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
  17. W.A. Kirk, The modulus of k-rotundity, Boll. Un. Mat. Ital. A, 7(2) (1988), 195-201.
  18. T.-C. Lim, On moduli of k-convexity, Abstr. Appl. Anal., 4(4) (1999), 243-247. https://doi.org/10.1155/S1085337599000202
  19. S. Saejung, On the modulus of U-convexity, Abstr. Appl. Anal., 2005(1) (2005), 59-66. https://doi.org/10.1155/AAA.2005.59
  20. S. Saejung, Sufficiant conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl., 330(1) (2007), 597-604. https://doi.org/10.1016/j.jmaa.2006.07.087
  21. S. Saejung and J. Gao, The n-Dimensional U-Convexity and Geometry of Banach Spaces, J. Fixed Point Theorey, 16 (2015), 381-392.
  22. S. Saejung and J. Gao, On the Banas-Hajnose-Wedrychowicz type modulus of convexity and fixed point property, Nonlinear Funct. Anal. Appl., 21(4) (2016), 717-725.
  23. E. Silverman, Definitions of Lebesgue area for surfaces in metric spaces, Rivista Mat. Univ. Parma, 2 (1951), 47-76.
  24. B. Sims, "Ultra"-techniques in Banach space theory. Queen's Papers in Pure and Applied Mathematics, 60. Queen's University, Kingston, ON, 1982.