Evolutionary Quasi-variational Hemivariational Inequalities: Existence and Parameter Identification

This paper is concerned with an evolutionary quasi-variational hemivariational inequality in which both the convex and nonconvex energy functionals depend on the unknown solution. The inequality serves as a direct problem of the inverse problem of parameters identification. Employing a fixed point argument and tools from nonlinear analysis, we establish the solvability and weak compactness of the solution set to the direct problem. Then, general existence and weak compactness results for the regularized optimization inverse problem have been proved. Moreover, we illustrate the applicability of the results by an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and nonmonotone boundary conditions and a state constraint.