The Existence Problems of Solutions for a Class of Differential Variational-Hemivariational Inequality Problems

In this work, we used reflexive Banach spaces to study the differential variational-hemivariational inequality problems with constraints. We established a sequence of perturbed differential variational-hemivariational inequality problems with perturbed constraints and penalty coefficients. Then, for each perturbed inequality, we proved the unique solvability and convergence of the solutions to the problems. Following that, we proposed a mathematical model for a viscoelastic rod in unilateral contact equilibrium, where the unknowns were the displacement field and the history of the deformation. We used the abstract penalty method in the analysis of this inequality and provided the corresponding mechanical interpretations.