EXISTENCE FOR A QUASISTATIC VARIATIONAL-HEMIVARIATIONAL INEQUALITY

This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migorski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded sub-differential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications.