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Transactions of the American Mathematical Society

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On some Banach space properties
sufficient for weak normal structure
and their permanence properties


Authors: Brailey Sims and Michael A. Smyth
Journal: Trans. Amer. Math. Soc. 351 (1999), 497-513
MSC (1991): Primary 47H09, 47H10, 46B20
DOI: https://doi.org/10.1090/S0002-9947-99-01862-0
MathSciNet review: 1401788
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.


References [Enhancements On Off] (What's this?)

  • 1. M. A. Akcoglu and L. Sucheston, On uniform monotonicity of norms and ergodic theorems in function spaces, Proc. Conf. Commemorating First Centennial Circ. Mat. Palermo, Rend. Circ. Mat. Palermo (2) Suppl. No. 8 (1985), 325-335. MR 88m:46029
  • 2. J. Barwise (editor), Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics 90, North-Holland, Amsterdam, 1977. MR 56:15351
  • 3. L. P. Belluce, W. A. Kirk and E. F. Steiner, Normal structure in Banach spaces, Pacific J. Math. 26 (1968), 433-440. MR 38:1501
  • 4. T. Dominguez Benavides, Some properties of the set and ball measures of non-compactness and applications, J. London Math. Soc. 34 (1986), 120-128. MR 87k:47124
  • 5. T. Dominguez Benavides, Weak uniform normal structure in direct sum spaces, Studia Math. 103 (1992), 283-290. MR 94c:46024
  • 6. W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427-436. MR 81m:46030
  • 7. D. van Dulst, Reflexive and superreflexive Banach spaces, Mathematical Centre Tracts, Vol. 102, Mathematische Centrum, Amsterdam, 1978. MR 80d:46019
  • 8. D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centres in Banach spaces with norms of type $KK$, Banach space theory and its applications, Springer Verlag Lecture Notes in Maths., 991, (1983), 35-43. MR 84i:46027
  • 9. D. van Dulst and V. de Valk, $(KK)$-properties, normal structure and fixed points of nonexpansive mappings in Orlicz sequence spaces, Can. J. Math 28 (1986), 728-750. MR 87i:46049
  • 10. K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, 1990. MR 92c:47070
  • 11. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain Journal of Math. 10 (1980), 743-749. MR 82b:46016
  • 12. A. Jimenez-Melado, Stability of weak normal structure in James quasi reflexive space, Bull. Austral. Math. Soc. 46 (1992), 367-372. MR 93m:46011
  • 13. D. Kutzarova and T. Landes, Nearly uniform convexity of infinite direct sums, Boll. Un. Mat. Ital. (A) (7) 9 (1995), 527-534. MR 97c:46012
  • 14. D. Kutzarova, E. Maluta, and S. Prus, Property $(\beta )$ implies normal structure of the dual space, Rend. Circ. Mat. Palermo (2) 41 (1992), 353-368. MR 94i:46020
  • 15. T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125-143. MR 86e:46014
  • 16. T. Landes, Normal structure and the sum-property, Pacific J. Math. 123 (1986), 127-147. MR 87h:46043
  • 17. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. A. M. S. 73 (1967), 591-597. MR 35:2183
  • 18. S. Prus, On Bynum's fixed point theorem, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 535-545. MR 91k:47146
  • 19. B. Sims and M. Smyth, On non-uniform conditions giving weak normal structure, Quaestiones Mathematicae 18 (1995), 9-19. MR 96e:46021
  • 20. K. Tan and H. Xu, On fixed point theorems of nonexpansive mappings in product spaces, Proc. A. M. S. 113 (1991), 983-989. MR 92e:47074
  • 21. D. Tingley, The normal structure of James quasireflexive space, Bull. Austral. Math. Soc. 42 (1990), 95-100. MR 91h:46038
  • 22. G. Zhang, Weakly convergent sequence coefficient of product space, Proc. A. M. S. 117 (1993), 637-643. MR 93d:46037
  • 23. W. Zhao, Remarks on various measures of noncompactness, J. Math. Anal. Appl. 174 (1993), 290-297. MR 94b:54007

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Additional Information

Brailey Sims
Affiliation: Department of Mathematics, University of Newcastle, Newcastle, NSW 2308, Australia
Email: bsims@maths.newcastle.edu.au

Michael A. Smyth
Affiliation: Department of Mathematics, University of Newcastle, Newcastle, NSW 2308, Australia

DOI: https://doi.org/10.1090/S0002-9947-99-01862-0
Received by editor(s): November 27, 1995
Article copyright: © Copyright 1999 American Mathematical Society

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