Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces

Author: Andreas H. Hamel
Journal: Proc. Amer. Math. Soc. 131 (2003), 3025-3038
MSC (2000): Primary 49J40, 46A03
Published electronically: April 30, 2003
MathSciNet review: 1993209
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Abstract: A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

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Additional Information

Andreas H. Hamel
Affiliation: Department of Mathematics and Computer Sciences, Martin-Luther-University Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06099 Halle, Germany

Keywords: Phelps' lemma, Ekeland's variational principle, Dane\u{s}' drop theorem, efficiency, locally convex space
Received by editor(s): May 17, 2001
Published electronically: April 30, 2003
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society