Gap functions and global error bounds for history-dependent variational-hemivariational inequalities

. This paper is devoted to a generalized time-dependent variational-hemivariational inequality with history-dependent operators. First, we introduce a new concept of gap functions to the time-dependent variational-hemivariational inequality under consideration. Then, we consider a regularized function, which is proved to be a gap function of the inequality problem, and establish several important properties to the regularized function. Furthermore, an global error bound to the time-dependent variational-hemivariational inequality, which implicitly depends on the regularized gap function, is obtained. Finally, a quasi-static contact problem with the constitutive law involving a convex subdifferential inclusion and long memory effect is studied as an illustrative application