Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

Abstract

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.