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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

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