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A converse to the Lions-Stampacchia theorem

Emil Ernst, Michel Théra (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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.

A converse to the Lions-Stampacchia Theorem

Emil Ernst, Michel Théra (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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.

A selection theorem of Helly type and its applications

Ehrhard Behrends, Kazimierz Nikodem (1995)

Studia Mathematica

We prove an abstract selection theorem for set-valued mappings with compact convex values in a normed space. Some special cases of this result as well as its applications to separation theory and Hyers-Ulam stability of affine functions are also given.

Diameter-preserving maps on various classes of function spaces

Bruce A. Barnes, Ashoke K. Roy (2002)

Studia Mathematica

Under some mild assumptions, non-linear diameter-preserving bijections between (vector-valued) function spaces are characterized with the help of a well-known theorem of Ulam and Mazur. A necessary and sufficient condition for the existence of a diameter-preserving bijection between function spaces in the complex scalar case is derived, and a complete description of such maps is given in several important cases.

Directions De Majoration D'une Fonction Quasiconvexe Et Applications

Amara, Charki (1998)

Serdica Mathematical Journal

We introduce the convex cone constituted by the directions of majoration of a quasiconvex function. This cone is used to formulate a qualification condition ensuring the epiconvergence of a sequence of general quasiconvex marginal functions in finite dimensional spaces.

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