Existence Results for Quasi-variational Inequalities with Applications to Radner Equilibrium Problems

Abstract

Local solutions for variational and quasi-variational inequalities are usually the best type of solutions that could practically be obtained when in case of lack of convexity or else when available numerical techniques are too limited for global solutions. Nevertheless, the analysis of such problems found in the literature seems to be very restricted to the global treatment. Motivated by this fact, in this work, we propose local solution concepts, study their interrelations and relations with global concepts and prove existence results as well as stability of local solution map of parametric variational inequalities.The key ingredient of our results is the new concept of local reproducibility of a set-valued map, which we introduce to explore such local solutions to quasivariational inequality problems. As a by-product, we obtain local solutions to quasi-optimization problems, bilevel quasi-optimization problems and Single-Leader-Multi-Follower games.

Existence Theorems on Quasi-variational Inequalities over Banach Spaces and its Applications to Time-dependent Pure Exchange Economy

“…Recently, Aussel et al [7] have studied a special type of QVI problem where some solutions can be obtained by deriving solutions of a corresponding VI problem. This special type of QVI problem is actually motivated by the Radner equilibrium problem [7,20] constructed from sequential trading game.…”

“…Recently, Aussel et al [7] have studied a special type of QVI problem where some solutions can be obtained by deriving solutions of a corresponding VI problem. This special type of QVI problem is actually motivated by the Radner equilibrium problem [7,20] constructed from sequential trading game. Indeed, this special type of QVI problem over Banach spaces is formed as: given setvalued mappings F : X → 2 X * and K : D → 2 X , and a function f : X → X * , the problem QVI(F, f, K) is to search for ( d, x) ∈ D × K( d) so that there is x * ∈ F (x) satisfying…”

“…The authors in [7] derived existence results for such QVI (2) problem over finite dimensional spaces under weaker semi continuity and quasimonotone assumption on F . As an application, they have derived the occurrence of Radner equilibrium [7] of certain sequential trading game. Few more works in this direction can be seen in Donato et al [17,19,20].…”

A Class of Quasi-Variational Inequalities with Unbounded Constraint Maps: Existence Results and Applications

“…al. [6] identified another class of QVI problems whose specific structure is beneficial in finding some of its solutions by solving a corresponding VI problem. In particular, the following class of QVI, denoted by QV I(A) is considered by them: for a given function g : R n → R m and the multi-valued maps G : R n ⇒ R n and M : P ⇒ R n where P = ∅ is a set in R m , the problem QV I(A) corresponds to, find (p, ȳ) ∈ P × M (p) such that ∃ ȳ * ∈ G(ȳ) satisfying g(ȳ), p − p + ȳ * , z − ȳ ≥ 0, for any (p, z) ∈ P × M (p).…”

“…It is noticeable that these problems were earlier considered by Donato-Milasi-Vitanza [13,14] in connection to the economic equilibrium problems. The primary motivation in [6] for studying these problems is to check the occurrence of Radner equilibrium for a sequential trading model with uncertainty. The authors in [6] proved the existence results for QV I(A) under the weak continuity and generalized monotonicity assumptions on the map G, by considering that the constraint map M admits bounded values.…”