Existence of Solutions for Non-monotone Variational Inequalities and Implications for Games

Abstract

In this paper, we study the existence of solutions in non-monotone variational inequalities (VIs) through the normal mapping properties. In particular, we provide sufficient conditions for the existence of solutions assuming that the normal mapping associated with the VI is norm coercive and its generalized Jacobian has certain properties, such as a full rank at points where the normal mapping is not zero. Then, we investigate sufficient conditions for the VI mapping and its Jacobian that ensure the generalized Jacobian of the normal mapping has full rank, such as the uniform P-function and the uniform P-matrix condition. Subsequently, we focus on VIs arising from games and interpret our results in a game setting and provide a sufficient condition for a game to have a Nash equilibrium. Through examples, we show that our sufficient conditions can be used to assert the existence of a solution to a VI, or a quasi-Nash in a game, while the existing results relying on the uniform P-function property or the P$_Υ$-matrix condition cannot be employed.