On Semicoercive Variational-Hemivariational Inequalities—Existence, Approximation, and Regularization

In this paper, we are concerned with semicoercive variational-hemivariational inequalities that encompass nonlinear semicoercive monotone variational inequalities (VIs) and pseudomonotone VIs in reflexive Banach spaces and hemivariational inequalities (HVIs) in function spaces. We present existence, approximation, and regularization results. Our approach to our existence result is based on recession arguments. We employ regularization techniques of nondifferentiable optimization to smooth the jumps in the hemivariational term. We treat nonconforming finite element approximation via Mosco convergence. As an example, we consider a semicoercive unilateral boundary value problem with nonmonotone boundary conditions that models a unilateral contact problem for a nonlinear elastic body under a nonmonotone friction law.