A converse to the Lions-Stampacchia Theorem

Emil Ernst; Michel Théra

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 4, page 810-817
  • ISSN: 1292-8119

Abstract

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In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits 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.

How to cite

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Ernst, Emil, and Théra, Michel. "A converse to the Lions-Stampacchia Theorem." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2008): 810-817. <http://eudml.org/doc/90938>.

@article{Ernst2008,
abstract = { In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits 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. },
author = {Ernst, Emil, Théra, Michel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Lions-Stampacchia Theorem; variational inequality; pseudo-monotone operator; Lions-Stampacchia theorem},
language = {eng},
month = {8},
number = {4},
pages = {810-817},
publisher = {EDP Sciences},
title = {A converse to the Lions-Stampacchia Theorem},
url = {http://eudml.org/doc/90938},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Ernst, Emil
AU - Théra, Michel
TI - A converse to the Lions-Stampacchia Theorem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/8//
PB - EDP Sciences
VL - 15
IS - 4
SP - 810
EP - 817
AB - In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits 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.
LA - eng
KW - Lions-Stampacchia Theorem; variational inequality; pseudo-monotone operator; Lions-Stampacchia theorem
UR - http://eudml.org/doc/90938
ER -

References

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  1. H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier18 (1968) 115–175.  
  2. G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).  
  3. G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema die Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei8 (1964) 91–140.  
  4. D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods, and Applications. Kluwer Academic Publishers (2003).  
  5. J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math.20 (1967) 493–519.  
  6. 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 Applications79, Springer-Verlag, New York (2005).  
  7. R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs49. American Mathematical Society (1997).  

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