ON A GENERALIZED GEOMETRIC CONSTANT AND SUFFICIENT CONDITIONS FOR NORMAL STRUCTURE IN BANACH SPACES

doi:10.1016/S0252-9602(17)30068-1

Abstract

Abstract:

In this paper,we introduce a new geometric constant C_NJ^(p)(a,X) of a Banach space X,which is closely related to the generalized von Neumann-Jordan constant and analyze some properties of the constant.Subsequently,we present several sufficient conditions for normal structure of a Banach space in terms of this new constant,the generalized James constant,the generalized García-Falset coefficient and the coefficient of weak orthogonality of Sims.Our main results of the paper generalize some known results in the recent literature.

Key words: uniform normal structure, generalized Garcí, a-Falset coefficient, coefficient of weak orthogonality, constant C_NJ^(p)(a,X), generalized James constant

Mina DINARVAND. ON A GENERALIZED GEOMETRIC CONSTANT AND SUFFICIENT CONDITIONS FOR NORMAL STRUCTURE IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2017, 37(5): 1209-1220.

Trendmd

References

[1] Brodskĭi M S, Mil'man D P. On the center of convex sets. (Russian) Doklady Akad Nauk SSSR (NS), 1948, 59:837-840
[2] Clarkson J A. The von Neumann-Jordan constant for the Lebesgue spaces. Ann of Math, 1937, 38(2):114-115
[3] Cui Y, Huang W, Hudzik H, Kaczmarek R. Generalized von Neumann-Jordan constant and its relationship to the fixed point property. Fixed Point Theory Appl, 2015, 2015:Article ID 40
[4] Dhompongsa S, Kaewkhao A. A note on properties that imply the weak fixed point property. Abstr Appl Anal, 2006, 2006:Article ID 34959
[5] Dhompongsa S, Kaewkhao A, Tasena S. On a generalized James constant. J Math Anal Appl, 2003, 285:419-435
[6] Dhompongsa S, Piraisangjun P, Saejung S. Generalized Jordan-von Neumann constants and uniform normal structure. Bull Aust Math Soc, 2003, 67:225-240
[7] Dinarvand M. On some Banach space properties sufficient for normal structure. Filomat, 2017, 31(5):1305-1315
[8] Dinarvand M. The James and von Neumann-Jordan type constants and uniform normal structure in Banach spaces. Int J Nonlinear Anal Appl, in press
[9] Domínguez-Benavides T. A geometrical coefficient implying the fixed point property and stability results. Houston J Math, 1996, 22(4):835-849
[10] Gao J, Lau K -S. On two classes of Banach spaces with uniform normal structure. Studia Math, 1991, 99(1):41-56
[11] Gao J, Saejung S. Normal structure and the generalized James and Zbăganu constants. Nonlinear Anal, 2009, 71:3047-3052
[12] Goebel K. Convexity of ball and fixed point theorems for mappings with nonexpansive square. Compositio Math, 1970, 22:269-274
[13] Goebel K, Kirk W A. Topics in Metric Fixed Point Theory. Cambridge:Cambridge University Press, 1990
[14] James R C. Uniformly non-square Banach spaces. Ann Math, 1964, 80(2):542-550
[15] Jiménez-Melado A, Llorens-Fuster E. The fixed point property for some uniformly nonsquare Banach spaces. Boll Unione Mat Ital Sez A, 1996, 10(7):587-595
[16] Jiménez-Melado A, Llorens-Fuster E, Saejung S. The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc Amer Math Soc, 2006, 134(2):355-364
[17] Khamsi M A. Uniform smoothness implies super-normal structure property. Nonlinear Anal, 1992, 19:1063-1069
[18] Kirk W A. A fixed point theorem for mappings which do not increase distances. Amer Math Monthly, 1965, 72:1004-1006
[19] Llorens-Fuster E. Zbăganu constant and normal structure. Fixed Point Theory, 2008, 9:159-172
[20] Mazcuñán-Navarro E M. Banach space properties sufficient for normal structure. J Math Anal Appl, 2008, 337:197-218
[21] Saejung S. On James and von Neumman-Jordan constants and sufficient conditions for the fixed point property. J Math Anal Appl, 2006, 323:1018-1024
[22] Saejung S. Sufficient conditions for uniform normal structure of Banach spaces and their duals. J Math Anal Appl 2007, 330:597-604
[23] Saejung S. The characteristic of convexity of a Banach space and normal structure. J Math Anal Appl, 2008, 337:123-129
[24] Sims B. Ultra-Techniques in Banach Space Theory. Queen's Papers in Pure and Applied Mathematics, Queen's University, 60, Kingston, 1982
[25] Sims B. Orthogonality and fixed points of nonexpansive maps. Proc Centre Math Anal Austral Nat Univ, Canberra, 1988:178-186
[26] Sims B. A class of spaces with weak normal structure. Bull Aust Math Soc, 1994, 50:523-528
[27] Wang X, Cui Y, Zhang C. The generalized von Neumann-Jordan constant and normal structure in Banach spaces. Ann Funct Anal, 2015, 6(4):206-214
[28] Zbăganu G. An equality of M. Rădulescu and S. Rădulescu which characterizes the inner product spaces. Rev Roumaine Math Pures Appl, 2002, 47(2):253-257