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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 $$y\in X$$”.

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