Existence of Solutions of Mixed Variational-like Inequalities and Gap Functions

In this work, a gap function and an extended gap function are introduced, which gives rise to an optimization problem formulation to mixed variational-like inequalities. Furthermore, a convex lower bound to an extended gap function for Stampachhia and Minty mixed variational-like inequality problems is developed. The results presented in this paper generalize some well-known results of several authors, which play a significant role in the theory of variational inequalities.