Generalized bi-quasi-variational inequalities

AbstractLet 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 → 2x and two multi-valued maps M, T: X → 2F, the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point ŷ ϵ X such that ŷ ϵ S(ŷ) and infw ϵ T(ȳ)Re〈ƒ − w, ŷ − x〉 ⩽ 0 for all x ϵ S(ŷ) and for all ƒ ϵ M(ŷ). 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

Existence of General Competitive Equilibria: A Variational Approach

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities