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

TL;DR

This paper establishes the existence of solutions for a broad class of quasi-variational inequalities with unbounded constraint maps, using coercivity conditions, and applies these results to economic equilibrium and game theory models.

Contribution

It introduces new existence results for quasi-variational inequalities with unbounded constraints, extending classical variational inequality theory.

Findings

Solutions exist under coercivity conditions

Applicability to economic equilibrium problems

Applicability to generalized Nash games

Abstract

The quasi-variational inequalities play a significant role in analyzing a wide range of real-world problems. However, these problems are more complicated to solve than variational inequalities as the constraint set is based on the current point. We study a class of quasi-variational inequality problems whose specific structure is beneficial in finding some of its solutions by solving a corresponding variational inequality problem. Based on the classical existence theorem for variational inequalities, our main results ensure the occurrence of solutions for the aforementioned class of quasi-variational inequalities in which the associated constraint maps are (possibly) unbounded. We employ a coercivity condition which plays a crucial role in obtaining these results. Finally, we apply our existence results to ensure the occurrence of equilibrium for the pure exchange economic problems and the jointly convex generalized Nash games.