Generalized quasi-variational inequalities in locally convex topological vector spaces

Abstract

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 ϵ E′ and x ϵ 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 y ̂ ϵ S( y ̂ ) and a point u ̂ ϵ T( y ̂ ) such that Re〈 u ̂ , y ̂ − x〉 ⩽ 0 for all x ϵ S( y ̂ ) . 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.