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A converse to the Lions–Stampacchia theorem. (English) Zbl 1176.47050

Summary: 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.

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H05 Monotone operators and generalizations
52A41 Convex functions and convex programs in convex geometry
39B82 Stability, separation, extension, and related topics for functional equations
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References:

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[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 Applications 79, Springer-Verlag, New York (2005). Zbl1093.01535 MR2160743 · Zbl 1093.01535
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