Lipschitz continuity of the solution maps to equilibrium problems

Abstract

This paper investigates the stability in the sense of Lipschitz continuity of the approximate solution maps to the parametric vector equilibrium problem in the normed spaces. More precisely, to achieve the Lipschitz continuity of the approximate solution maps for this problem, we used the Gerstewitz nonlinear scalar function (a very useful tool in studying properties solutions related to optimization problems) together with assumptions about the relaxed conditions related to concavity properties of the objective function. We also give an example showing that this property is weaker than the cone concavity of the vector-valued map. Besides, the Lipschitz continuity and the uniformly bounded diameter of the constrained map are both used. The approach and obtained results on Lipschitz continuity for this problem are new and different from the existing ones.