Generalized quasi-variational inequalities in locally convex topological vector spaces

Let E be a Hausdorff topological vector space and X ? E an arbitrary nonempty set. Denote by E′ the dual space of E and the pairing between E′ and E by 〈w, x〉 for w?E′ and x?E. Given a point-to-set map S: X → 2^X and a point-to-set map T: X → 2^E′, the generalized quasi-variational inequality problem (GQVI) is to find a point $\text{y}\text{?}\text{? S(}\text{y}\text{?}\text{)}$ and a point $\text{u}\text{?}\text{? T(}\text{y}\text{?}\text{)}$ such that $\text{Re}\text{〈}\text{u}\text{?}\text{,}\text{y}\text{?}\text{? x〉 ? 0}$ for all $\text{x ? S(}\text{y}\text{?}\text{)}$. By using the Ky Fan minimax principle or its generalized version as a tool, some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces are obtained which include a fixed point theorem due to Ky Fan and I. L. Glicksberg, and two different multivalued versions of the Hartman-Stampacchia variational inequality.