Generalized bi-quasi-variational inequalities

Abstract

Let E, F be Hausdorff topological vector spaces over the field Φ (which is either the real field or the complex field), let 〈 , 〉: F × E → Φ be a bilinear functional, and let X be a non-empty subset of E. Given a multi-valued map S: X → 2 x and two multi-valued maps M, T: X → 2 F , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point y ̂ ϵ X such that y ̂ ϵ S( y ̂ ) and inf w ϵ T( y ̄ ) Re〈ƒ − w, y ̂ − x〉 ⩽ 0 for all x ϵ S( y ̂ ) and for all ƒ ϵ M( y ̂ ) . In this paper two general existence theorems on solutions of GBQVIs are obtained which simultaneously unify, sharpen, and extend existence theorems for multi-valued versions of Hartman-Stampacchia variational inequalities proved by Browder and by Shih and Tan, variational inequalities due to Browder, existence theorems for generalized quasi-variational inequalities achieved by Shih and Tan, theorems for monotone operators obtained by Debrunner and Flor, Fan, and Browder, and the Fan-Glicksberg fixed-point theorem.