Variational inequalities for functions on convex sets (Q791860)

A class of abstract variational inequalities is considered in an abstract setting, namely in a Hausdorff linear topological space Y. The essential results of this note are contained in three theorems. Most important of them seems to be the following one: Theorem. Let K be a compact, convex subset of Y. Then (1), (2), (3) and (4) below are equivalent. (1) There exists \(w\in K\) such that for all \(x\in Y:\) Aw(w)\(\leq Aw(x).\) (2) There exists \(w\in K\) such that for all \(x\in Y:\) Ax(w)\(\leq Ax(x).\) (3) For all \(x\in Y\) there exists \(w\in K\) such that Aw(w)\(\leq Aw(x).\) (4) For all \(x\in Y\) there exists \(w\in K\) such that Ax(w)\(\leq Ax(x).\) Here Ax is, for all \(x\in Y\), a real convex lower semicontinuous function on Y, such that if \(x\to y\) along a line segment then A\(x\to Ay\) pointwise on Y. No examples for the application of the general results to specific problems are given. For the reader interested in this paper the following contributions will also be useful: \textit{N. Hirano}, \textit{W. Takahashi} [Proc. Am. Math. Soc. 80, 647-650 (1980; Zbl 0471.49011)], \textit{W. Takahashi} [Southeast Asian Bull. Math. 4, 59-85 (1980; Zbl 0487.47037)].