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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 $f:X×X\to ℝ$ be a mapping such that:

(1) for each fixed $y\in X$, $f\left(·,y\right):X\to ℝ$ is weakly usc on $X$.

(2) there exists a real $c$ such that

(i) for each $x\in X$ and $t, the set $\left\{y\in X:f\left(x,y\right)\le t\right\}$ is convex,

(ii) for each $x\in X$, $f\left(x,x\right)\ge c$,

(iii) for a particular ${y}_{0}\in X$, the set $\left\{x\in X:f,{y}_{0}\right)\ge c\right\}$ is a bounded subset of $E$.

Then there exists an ${x}_{0}\in X$ such that $f\left({x}_{0},y\right)\ge c$ for all $y\in X$”.

MSC:
 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)