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Equicontinuity of Families of Convex and Concave-Convex Operators

Published online by Cambridge University Press:  20 November 2018

Mohamed Jouak  and
Lionel Thibault
Mohamed Jouak
Affiliation:
Université de Pau, Pau, France
Lionel Thibault
Affiliation:
Université de Pau, Pau, France
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J. M. Borwein has given in [1] a practical necessary and sufficient condition for a convex operator to be continuous at some point. Indeed J. M. Borwein has proved in his paper that a convex operator with values in an order topological vector space F (with normal positive cone F+) is continuous at some point if and only if it is bounded from above by a mapping which is continuous at this point. This result extends a previous one by M. Valadier in [16] asserting that a convex operator is continuous at a point whenever it is bounded from above by an element in F on a neighbourhood of the concerned point. Note that Valadier's result is necessary if and only if the topological interior of F+ is nonempty. Obviously both results above are generalizations of the classical one about real-valued convex functions formulated in this context exactly as Valadier's result (see for example [5]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 5 , 01 October 1984 , pp. 883 - 898
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Borwcin, J. M., Continuity and differentiability properties of convex operators, Proc. London Math. Soc. 44 (1982), 420–444.Google Scholar
2. Borwcin, J. M., A Lagrange multiplier theorem and a sandwich theorem for convex relations. Math. Scand 48 (1981), 189–204.Google Scholar
3. Borwcin, J. M., Convex relations in analysis and optimization in generalized concavity in optimization and economics (Academic Press, London, 1981), 335–371.Google Scholar
4. Borwein, J. M., Penot, J. P. and Thera, M., Conjugate vector-valued convex mappings, J. Math Anal. Appl., to appear.Google Scholar
5. Bourbaki, N., Espaces vectoriels topologiques (Hermann, Paris, 1964).Google Scholar
6. Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions. Lecture Notes in Mathematics 580 (Springer-Verlag, Berlin, 1977).CrossRefGoogle Scholar
7. Dolecki, S., Salinetti, G. and Wets, R. J. B., Convergence of functions: equisemicontinuity, Trans. Amer. Math. Soc, to appear.Google Scholar
8. Jouak, M. and Thibault, L., Directional derivatives and almost everywhere differentiability oj biconvex and concave-convex operators, to appear.Google Scholar
9. Jouak, M. and Thibault, L., Monotomie généralisée et sousdifférentiels de jonctions convexes vectorielles, to appear.Google Scholar
10. Penot, J. P. and Thera, M., Semicontinuous mappings in general topology. Arch. Math. 38 (1982), 158–166.Google Scholar
11. Peressini, A. L., Ordered topological vector spaces (Harper and Row, New-York, 1967).Google Scholar
12. Rockafellar, R. T., Convex analysis (Princeton Univ. Press, Princeton, 1970).CrossRefGoogle Scholar
13. Thera, M., Etude des fonctions convexes vectorielles semicontinues, Thèse de Spécialité, Pau (1978).Google Scholar
14. Thibault, L., Subdifferentials of compactly lipschitzian vector-valued Junctions, Ann. Math. Pura Appl. 125 (1980), 157–192.Google Scholar
15. Thibault, L., Continuity of measurable convex and biconvex operators, to appear.CrossRefGoogle Scholar
16. Valadier, M., Sous-différentiabilité des fonctions convexes à valeurs dans un espace vectoriel ordonne, Math. Scand. 30 (1972), 65–74.Google Scholar