Differential Evolution Hemivariational Inequalities with Anti-periodic Conditions

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality (DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space. First, by applying surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient, we show the nonempty of the solution set for the parabolic hemivariational inequality. Then, some topological properties of the solution set are established such as boundedness, closedness and convexity. Furthermore, we explore the upper semicontinuity of the solution mapping. Finally, we prove the solution set of the system (DEHVI) is nonempty and the set of all trajectories of (DEHVI) is weakly compact in C(I, X).