New existence results for evolutionary quasi-hemi-variational inequalities and their optimal control

Abstract

This paper investigates evolutionary quasi-variational inequalities that involve multi-valued monotone and pseudo-monotone maps. The purpose of including the pseudo-monotone map is to encompass generalized derivatives of certain nonconvex locally Lipschitz functionals as a specific instance. Consequently, the considered evolutionary quasi-variational inequalities subsume evolutionary quasi-hemi-variational inequalities as a particular case. We present an existence result wherein the key novelty lies in the variational selection that separates the monotone and pseudo-monotone components, thus eliminating the conventional assumption that their sum is monotone. While we continue to employ the Minty formulation, its necessity is limited to evolutionary monotone variational inequalities alone. Furthermore, we offer new applications to optimal control problems governed by evolutionary quasi-variational inequalities.