*(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 $y\in X$, $f(\xb7,y):X\to \mathbb{R}$ is weakly usc on $X$.

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

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

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

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

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