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.