New Existence Results for Evolutionary Quasi-Hemi-Variational Inequalities with Semi-monotone Maps and Applications to Optimal Control Problems

The paper presents new results for the existence of solutions for an evolutionary quasi-hemi-variational inequality, which involves a semi-monotone and a pseudo-monotone map. A crucial aspect of this research lies in the utilization of variational selection to disentangle the monotonic and pseudo-monotonic components. By employing this technique, we can bypass the commonly made assumption that the sum of the two maps must be monotone. Consequently, we establish various new existence results for evolutionary quasi-hemi-variational inequalities. Furthermore, we demonstrate applications of these results in solving optimal control problems.