A generalized penalty method for a new class of differential inequality system

The primary objective of this paper is to study a nonlinear system involving a parabolic vari-ational inequality, a history-dependent hemivariational inequality and a differential equation constraints in a Banach space. First, we derive a unique solvability theorem for such problem under some mild hypotheses. Second, we construct a penalized problem for such nonlinear system, and show the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the sequence of approximate solution to the solution of the original system when the penalty parameter converges to zero. Finally, these results are applied to a quasistatic elastic frictional contact problem with heat equation with memory, and damage.

Automated summary

This paper studies a nonlinear system involving a parabolic variational inequality, a history-dependent hemivariational inequality, and differential equation constraints in a Banach space. The unique solvability theorem is derived, and a penalized problem is constructed to obtain an approximating sequence for the nonlinear system. Moreover, the strong convergence of the sequence of approximate solution to the solution of the original system is proved when the penalty parameter converges to zero. Finally, these results are applied to a quasistatic elastic frictional contact problem with heat equation with memory and damage.