Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T16:51:15.894Z Has data issue: false hasContentIssue false
  • English
  • Français

On Modulus of Noncompact Convexity and Its Properties

Published online by Cambridge University Press:  20 November 2018

Józef Banaś
Józef Banaś*
Affiliation:
Department of Mathematics Technical University of Rzeszów 35-084 Rzeszów, Poznańska 2, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we prove some properties of the so-called modulus of noncompact convexity. This notion was recently introduced by K. Goebel and T. Sȩkowski [6] and it appears to be an interesting and useful generalization of the classical Clarkson modulus of convexity. We extend the results obtained in [6] showing that the modulus of noncompact convexity is continuous and has some extra properties in reflexive Banach spaces. The properties applicable in the fixed point theory are also stated.

Keywords

46B20
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 2 , 01 June 1987 , pp. 186 - 192
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Banaś, J. and Goebel, K., Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics , Marcel Dekker, Vol. 60, New York, 1980.Google Scholar
2. Clarkson, J.A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), pp. 396–414.Google Scholar
3. Day, M.M., Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), pp. 313–317.Google Scholar
4. Enflo, P., Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), pp. 281–288.Google Scholar
5. Goebel, K. and Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, New York, Basel, 1984.Google Scholar
6. Goebel, K. and Sȩkowski, T., The modulus of noncompact convexity, Ann. Univ. Mariae Curie-Skłodowska, Sect. A (preprint).Google Scholar
7. Kirk, W.A., A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72(1965), pp. 1004–1006.Google Scholar
8. Köthe, G., Topological vector spaces I, Springer Verlag, 1969.Google Scholar
9. Opial, Z., Lecture notes on nonexpansive and monotone mappings in Banach spaces, Center for Dynamical Systems , Brown University, 1967.Google Scholar
10. Radström, J., An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), pp. 165–169.Google Scholar