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Zbl 1034.49005
Isac, G.; Sehgal, V. M.; Singh, S. P.
An alternate version of a variational inequality.
(English)
[J] Indian J. Math. 41, No. 1, 25-31 (1999). ISSN 0019-5324

From the text: The purpose of this paper is to give some other alternate version of a result, concerning a variational inequality due to {\it G. Allen} [J. Math. Anal. Appl. 58, 1--10 (1977; Zbl 0383.49005)].\par The following theorem is the main result of this paper.\par Theorem. Let $X$ be a closed nonempty subset of a locally convex semi-reflexive topological vector space $E$ and let $f:X\times X\to{\Bbb R}$ be a mapping such that:\par (1) for each fixed $y\in X$, $f(\cdot,y):X\to{\Bbb R}$ is weakly usc on $X$.\par (2) there exists a real $c$ such that\par (i) for each $x\in X$ and $t<c$, the set $\{y\in X:f(x,y)\le t\}$ is convex,\par (ii) for each $x\in X$, $f(x,x)\ge c$,\par (iii) for a particular $y_0\in X$, the set $\{x\in X:f,y_0)\ge c\}$ is a bounded subset of $E$.\par Then there exists an $x_0\in X$ such that $f(x_0,y)\ge c$ for all $y\in X$''.
MSC 2000:
*49J40 Variational methods including variational inequalities
90C33 Complementarity problems

Keywords: Ekeland's principle; KKM-mapping; variational inequality; complementarity

Citations: Zbl 0383.49005

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