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An alternate version of a variational inequality. (English) Zbl 1034.49005
From the text: “The purpose of this paper is to give some other alternate version of a result, concerning a variational inequality due to G. Allen [J. Math. Anal. Appl. 58, 1–10 (1977; Zbl 0383.49005)].
The following theorem is the main result of this paper.
Theorem. Let \(X\) be a closed nonempty subset of a locally convex semi-reflexive topological vector space \(E\) and let \(f:X\times X\to{\mathbb R}\) be a mapping such that:
(1) for each fixed \(y\in X\), \(f(\cdot,y):X\to{\mathbb R}\) is weakly usc on \(X\).
(2) there exists a real \(c\) such that
(i) for each \(x\in X\) and \(t<c\), the set \(\{y\in X:f(x,y)\leq t\}\) is convex,
(ii) for each \(x\in X\), \(f(x,x)\geq c\),
(iii) for a particular \(y_0\in X\), the set \(\{x\in X:f,y_0)\geq c\}\) is a bounded subset of \(E\).
Then there exists an \(x_0\in X\) such that \(f(x_0,y)\geq c\) for all \(y\in X\)”.

49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)