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A converse to the Lions-Stampacchia Theorem

Published online by Cambridge University Press:  20 August 2008

Emil Ernst  and
Michel Théra
Emil Ernst
Affiliation:
Aix-Marseille Univ, UMR6632, Marseille, 13397, France. Emil.Ernst@univ-cezanne.fr
Michel Théra
Affiliation:
XLIM (UMR-CNRS ) and Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France. michel.thera@unilim.fr
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Abstract

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert spaceadmits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

Keywords

Lions-Stampacchia Theoremvariational inequalitypseudo-monotone operator
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 810 - 817
Copyright
© EDP Sciences, SMAI, 2008

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References

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Lions, J.-L. and Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math. 20 (1967) 493–519. CrossRef
J.-L. Lions, E. Magenes, O.G. Mancino and S. Mazzone, Variational Analysis and Applications, in Proceedings of the 38th Conference of the School of Mathematics “G. Stampacchia", in memory of Stampacchia and J.-L. Lions, Erice, June 20–July 1st 2003, F. Giannessi and A. Maugeri Eds., Nonconvex Optimization and its Applications 79, Springer-Verlag, New York (2005).
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