Weak uniform normal structure and iterative fixed points of nonexpansive mappings

T. Domínguez Benavides; G. López Acedo; Hong Xu

Colloquium Mathematicae (1995)

  • Volume: 68, Issue: 1, page 17-23
  • ISSN: 0010-1354

Abstract

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This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.

How to cite

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Domínguez Benavides, T., López Acedo, G., and Xu, Hong. "Weak uniform normal structure and iterative fixed points of nonexpansive mappings." Colloquium Mathematicae 68.1 (1995): 17-23. <http://eudml.org/doc/210288>.

@article{DomínguezBenavides1995,
abstract = {This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.},
author = {Domínguez Benavides, T., López Acedo, G., Xu, Hong},
journal = {Colloquium Mathematicae},
keywords = {nonexpansive mapping; iterative fixed point; geometrical coefficients of Banach spaces; James' quasi-reflexive space; weak uniform normal structure; iterative fixed points of nonexpansive mappings; weakly convergent sequence coefficient; James' quasi-reflexive spaces},
language = {eng},
number = {1},
pages = {17-23},
title = {Weak uniform normal structure and iterative fixed points of nonexpansive mappings},
url = {http://eudml.org/doc/210288},
volume = {68},
year = {1995},
}

TY - JOUR
AU - Domínguez Benavides, T.
AU - López Acedo, G.
AU - Xu, Hong
TI - Weak uniform normal structure and iterative fixed points of nonexpansive mappings
JO - Colloquium Mathematicae
PY - 1995
VL - 68
IS - 1
SP - 17
EP - 23
AB - This paper is concerned with weak uniform normal structure and iterative fixed points of nonexpansive mappings. Precisely, in Section 1, we show that the geometrical coefficient β(X) for a Banach space X recently introduced by Jimenez-Melado [8] is exactly the weakly convergent sequence coefficient WCS(X) introduced by Bynum [1] in 1980. We then show in Section 2 that all kinds of James' quasi-reflexive spaces have weak uniform normal structure. Finally, in Section 3, we show that in a space X with weak uniform normal structure, every nonexpansive self-mapping defined on a weakly sequentially compact convex subset of X admits an iterative fixed point.
LA - eng
KW - nonexpansive mapping; iterative fixed point; geometrical coefficients of Banach spaces; James' quasi-reflexive space; weak uniform normal structure; iterative fixed points of nonexpansive mappings; weakly convergent sequence coefficient; James' quasi-reflexive spaces
UR - http://eudml.org/doc/210288
ER -

References

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  1. [1] W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427-436. Zbl0442.46018
  2. [2] T. Domínguez Benavides, Weak uniform normal structure in direct-sum spaces, Studia Math. 103 (1992), 283-290. Zbl0810.46015
  3. [3] T. Domínguez Benavides, Some properties of the set and ball measures of noncompactness and applications, J. London Math. Soc. 34 (1986), 120-128. Zbl0578.47045
  4. [4] T. Domínguez Benavides and G. López Acedo, Lower bounds for normal structure coefficients, Proc. Roy. Soc. Edinburgh 121A (1992), 245-252. Zbl0787.46010
  5. [5] M. Edelstein and R. C. O'Brien, Nonexpansive mappings, asymptotic regularity, and successive approximations, J. London Math. Soc. 17 (1978), 547-554. Zbl0421.47031
  6. [6] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. Zbl0708.47031
  7. [7] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71. Zbl0352.47024
  8. [8] A. Jimenez-Melado, Stability of weak normal structure in James quasi reflexive space, Bull. Austral. Math. Soc. 46 (1992), 367-372. 
  9. [9] M. A. Khamsi, James quasi reflexive space has the fixed point property, ibid. 39 (1989), 25-30. Zbl0672.47045
  10. [10] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. Zbl0141.32402
  11. [11] E. Maluta, Uniformly normal structure and related coefficients for Banach spaces, Pacific J. Math. 111 (1984), 357-369. Zbl0495.46012
  12. [12] P. M. Soardi, Schauder bases and fixed points of nonexpansive mappings, Pacific J. Math. 101 (1982), 193-198. Zbl0457.47046

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