Weighted quasi-variational inequalities and constrained Nash equilibrium problems

The weighted quasi-variational inequalities over product of sets (for short, WQVIP) and system of weighted quasi-variational inequalities (for short, SWQVI) are introduced. It is shown that these two problems are equivalent. A relationship between SWQVI and system of vector quasi-variational inequalities is given. The concept of normalized solutions of WQVIP and SWQVI is introduced. A relationship between solution (respectively, normalized solution) of SWQVI and solution of weighted constrained Nash equilibrium problem (respectively, normalized weight Nash equilibrium) is also given. The scalar quasi-equilibrium problem (for short, QEP), which includes WQVIP as a particular case, is also considered. By introducing the concept of densely pseudomonotonicity of bifunctions, the existence of a solution of QEP is established. As a consequence, existence results for solutions of WQVIP and constrained Nash equilibrium problems for vector valued functions are derived.