Existence and uniqueness of the solution to a new class of evolutionary variational hemivariational inequalities

Abstract

This paper is concerned with an evolutionary variational hemivariational inequality which is considered in the form of a nonlinear evolution inclusion. In the inclusion, both the convex subdifferential and Clarke subdifferential are related to the time derivative of the unknown function. In addition, the convex subdifferential operator is unbounded and thus the Signorini case is included. Due to these features, the existing surjectivity theorems for evolution inclusions are not applicable. Instead, the Rothe method based on the temporal discretization strategy is used to study the solvability of this new variational hemivariational inequality. We first show the existence of solutions to the discrete stationary problem. Then we establish a convergence result of the semidiscrete scheme and prove the existence and uniqueness of the solution to the inclusion. Moreover, we show the existence and uniqueness of the solution to the original variational hemivariational inequality. Finally, an example is given to illustrate the abstract result.