SOME CHARACTERIZATIONS OF THE APPROXIMATE SOLUTIONS TO GENERALIZED VECTOR EQUILIBRIUM PROBLEMS

In this paper, a scalarization result and a density theorem concerned with the sets of weakly efficient and efficient approximate solutions to a generalized vector equilibrium problem are given, respectively. By using the scalarization result and the density theorem, the connectedness of the sets of weakly efficient and efficient approximate solutions to the generalized vector equilibrium problem are established without the assumptions of monotonicity and compactness. The lower semicontinuity of weakly efficient and efficient approximate solution mappings to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are obtained without the assumptions of monotonicity and compactness. Furthermore, the upper semicontinuity of weakly efficient approximate solution mapping and the Hausdorff upper semicontinuity of efficient approximate solution mapping to the parametric generalized vector equilibrium problem with perturbing both the objective mapping and the feasible region are also given under some suitable conditions.