New generalizations of Browder's variational inequalities and the Ky Fan minimax inequality (Q1368470)

This paper established the following generalized version of Browder's variational inequality theorem. Theorem 1. Let \((X,\{F_A\})\) be a compact \(H\)-space, \(E\) a locally convex Hausdorff topological vector space, \(Y\) a compact convex subset of \(E\) and \(T:X\to 2^Y\) an u.s.c. multimapping with nonempty compact convex values, \(\varphi:X\times Y\times X\to\mathbb{R}\) be u.s.c. such that (i) \(\varphi(x,y,z)\) is quasiconcave in \(y\) and \(H\)-quasiconvex in \(z\), resp., (ii) for each \(x\in X\), there exists a \(y\in T(x)\) such that \(\varphi(x,y,x)\geq 0\). Then there exist \(\overline x\in X\), \(\overline y\in T(\overline x)\) such that \(\varphi(\overline x,\overline y,x)\geq 0\) for all \(x\in X\). If \(X\) is a nonempty compact convex subset of a Hausdorff locally convex space \(E\), \(Y= E^*\) and \(\varphi(x,y,z)= \langle y,x-z\rangle\) then Theorem 1 reduces to Theorem 1 of Browder (1968). The paper also established a new generalization of the Ky Fan minimax inequality. This result also includes a result of Yen (1981) as a special case.