Levitin–Polyak well‐posedness by perturbations for systems of set‐valued vector quasi‐equilibrium problems

This paper is devoted to the Levitin–Polyak well‐posedness by perturbations for a class of general systems of set‐valued vector quasi‐equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set‐valued vector quasi‐equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well‐posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well‐posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well‐posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.