Existence of Projected solutions for a quasi-equilibrium problem with applications

This article studies a quasi-equilibrium problem (QEP, for short) in an infinite dimensional Banach space X. Under very mild assumptions, we establish nonemptiness and compactness of projected solution set to (QEP), both in the cases when X is reflexive or not. The main strategy is based on the Michael selection theorem, the Schauder fixed-point theorem, an approximating technique and the concept of approximatively compactness. As application of the obtained results, we focus on three problems with non-self and implicit constraints which are, respectively, a quasi-variational-hemivariational inequality, a quasi-optimization problem, and a generalized Nash equilibrium problem.