Generalized von Neumann-Jordan constant and its relationship to the fixed point property

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• 作者：Yunan Cui (1)
  Wan Huang (1)
  Henryk Hudzik (2)
  Rados艂aw Kaczmarek (2)
  
  1. Department of Mathematics ; Harbin University of Science and Technology ; Harbin ; 150080 ; P.R. China
  2. Department of Mathematics ; Faculty of Mathematics and Computer Science ; Adam Mickiewicz University in Pozna艅 ; Pozna艅 ; Poland
  
• 关键词：46B20 ; 46E30 ; the generalized von Neumann ; Jordan constant ; James constant ; uniform non ; squareness ; Lebesgue space ; normal structure ; weak fixed point property ; fixed point property
• 刊名：Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：2015
• 期：1
• 全文大小：1,183 KB
• 参考文献：1. Jordan, P, Neumann, J (1935) On inner products in linear metric spaces. Ann. Math. 36: pp. 719-723 CrossRef
  2. Zuo, Z, Cui, Y (2009) Some sufficient conditions for fixed points of multivalued nonexpansive mappings. Fixed Point Theory Appl. 2009: CrossRef
  3. Clarkson, JA (1937) The von Neumann-Jordan constant for the Lebesgue space. Ann. Math. 38: pp. 114-115 CrossRef
  4. Kato, M, Maligranda, L, Takahashi, Y (2001) On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces. Stud. Math. 144: pp. 275-295 CrossRef
  5. Kato, M, Maligranda, L (2001) On James and Jordan-von Neumann constants of Lorentz sequence spaces. J. Math. Anal. Appl. 258: pp. 457-465 CrossRef
  6. Takahashi, Y, Kato, M (1998) Von Neumann-Jordan constant and uniformly non-square Banach spaces. Nihonkai Math. J. 9: pp. 155-169
  7. Yang, C (2014) Jordan-von Neumann constant for Bana艣-Fr膮czek space. Banach J. Math. Anal. 8: pp. 185-192 CrossRef
  8. Garc铆a-Falset, J, Llorens-Fuster, E, Mazcu艅an-Navarro, EM (2006) Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal. 233: pp. 494-514 CrossRef
  9. Yang, C, Wang, F (2006) On a new geometric constant related to the von Neumann-Jordan constant. J. Math. Anal. Appl. 324: pp. 555-565 CrossRef
  10. Lindenstrauss, J (1963) On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J. 10: pp. 241-252 CrossRef
  11. Day, MM (1944) Uniform convexity in factor and conjugate spaces. Ann. Math. 45: pp. 375-385 CrossRef
  12. Takahashi, Y, Hashimoto, K, Kato, M (2002) On sharp uniform convexity, smoothness and strong type, cotype inequalities. J.聽Nonlinear Convex Anal. 3: pp. 267-281
  13. Zuo, Z, Cui, Y (2009) A coefficient related to some geometric properties of a Banach space. J. Inequal. Appl. 2009: CrossRef
  14. Gao, J (2005) Normal structure and Pythagorean approach in Banach space. Period. Math. Hung. 51: pp. 19-30 CrossRef
  15. Jim茅nez-Melado, A, Llorens-Fuster, E (1996) The fixed point property for some uniformly nonsquare Banach spaces. Boll. Unione Mat. Ital., A (7) 10: pp. 587-595
  16. Kirk, WA (1965) A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 72: pp. 1004-1006 CrossRef
  17. Goebel, K, Kirk, WA (1990) Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge CrossRef
  18. Kirk, WA, Sims, B (2001) Handbook of Metric Fixed Point Theory. Kluwer Academic, Dordrecht CrossRef
  19. Kato, M, Takahashi, Y (1997) Type, cotype constants and Clarkson鈥檚 inequalities for Banach spaces. Math. Nachr. 186: pp. 187-195 CrossRef
  20. Jim茅nez-Melado, A, Llorens-Fuster, E, Saejung, S (2005) The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces. Proc. Am. Math. Soc. 134: pp. 355-364 CrossRef
  21. James, RC (1964) Uniformly non-square Banach spaces. Ann. Math. 80: pp. 542-550 CrossRef
  
• 刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
• 出版者：Springer International Publishing
• ISSN：1687-1812

文摘

We introduce a new geometric constant \(C_{NJ}^{(p)}(X)\) for a Banach space X, called a generalized von Neumann-Jordan constant. Next, it is shown that \(1\leq C_{NJ}^{(p)}(X)\leq2\) for any Banach space X and that the right hand side inequality is sharp if and only if X is uniformly non-square. Moreover, a relationship between the James constant \(J(X)\) and \(C_{NJ}^{(p)}(X)\) is presented. Finally, the generalized von Neumann-Jordan constant of the Lebesgue space \(L_{r}([0,1])\) is calculated and a relationship between \(C_{NJ}^{(p)}(X)\) and the fixed point property is found.