Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-06T11:07:43.408Z Has data issue: false hasContentIssue false

The implications for differentiability of a weak index of non-compactness

Published online by Cambridge University Press:  17 April 2009

John R. Giles
Affiliation:
Department of Mathematics, The University of Newcastle, New South Wales 2308, Australia
Warren B. Moors
Affiliation:
Department of Mathematics, The University of Newcastle, New South Wales 2308, Australia
Rights & Permissions [Opens in a new window]

Extract

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 a recent paper the authors showed that certain set-valued mappings from a Baire space into subsets of a Banach space which have a continuity property defined in terms of Kuratowski's index of non-compactness have inherent single-valued properties. Here we generalise the continuity property to one defined in terms of a weak index of non-compactness and we show that this wider class of set-valued mappings also has significant implications for the differentiability of convex functions on Banach spaces.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Banás, Józef and Rivero, Jesus, ‘On measures of weak non-compactness’, Ann. Mat. Pura Appl. (4) 151 (1988), 213224.CrossRefGoogle Scholar
[2]Bourgain, J., ‘Strongly exposed points in weakly compact convex sets in Banach spaces’, Proc. Amer. Math. Soc. 58 (1976), 197200.CrossRefGoogle Scholar
[3]Day, M.M., Normed linear spaces 3rd ed. (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[4]de Blasi, F.S., ‘On a property of the unit sphere in a Banach Space’, Bull. Math. Soc. Sci. Math. R.S. Roumaine (N.S.) 21 (1977), 259262.Google Scholar
[5]Giles, J.R., Gregory, D.A. and Sims, B., ‘Geometrical implications of upper semi-continuity of the duality mapping on a Banach space’, Pacific J. Math. 79 (1978), 99109.CrossRefGoogle Scholar
[6]Giles, J.R. and Kutzarova, D.N., ‘Characterisation of drop and weak drop properties for closed bounded convex sets’, Bull. Austral. Math. Soc. 43 (1991), 377385.CrossRefGoogle Scholar
[7]Giles, J.R. and Moors, W.B., ‘A continuity property related to Kuratowski's index of non-compactness, its relevance to the drop property and its implications for differentiability theory’, J. Math. Anal. Appl. (to appear).Google Scholar
[8]Jokl, L., ‘Minimal convex valued weak* usco correspondences and the Radon-Nikodym property’, Comm. Math. Univ. Carolinae 28 (1987), 353376.Google Scholar
[9]Kenderov, P.S. and Giles, J.R., ‘On the structure of Banach spaces with Mazur's intersection property’, Math. Ann. 291 (1991), 463473.CrossRefGoogle Scholar
[10]Kutzarova, D.N., ‘On the drop property of convex sets in Banach spaces’, in Constructive theory of functions, 1987 (Sofia, 1988), pp. 283287.Google Scholar
[11]Kutzarova, D.N. and Papini, P.L., ‘On some properties concerning convex sets and weak convergence’. Preprint.Google Scholar
[12]Namioka, I., ‘Separate continuity and joint continuity’, Pacific J. Math. 51 (1974), 515531.CrossRefGoogle Scholar
[13]Namioka, I. and Phelps, R.R., ‘Banach spaces which are Asplund spaces’, Duke Math. J. 42 (1975), 735750.CrossRefGoogle Scholar
[14]Phelps, R.R., Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1989).CrossRefGoogle Scholar
[15]Phelps, R.R., ‘Weak* support poins of convex sets in E*’, Israel J. Math. 2 (1964), 177182.CrossRefGoogle Scholar
[16]Troyanski, S.L., ‘On a property of the norm which is close to local uniform rotundity’, Math. Ann. 271 (1985), 305314.CrossRefGoogle Scholar
[17]Troyanski, S.L., ‘On some generalisations of denting points’. Preprint.Google Scholar