We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence of generalized penalty problems, governed by a parameter ${\lambda }_n$ and an operator $P_n$. We prove the unique solvability of the penalty problems as well as the convergence of corresponding solutions sequence to the solution of original problem. These results extend the previous results in Sofonea et al. (2018) and Xiao and Sofonea (2019). Finally, we illustrate them in the study of a history-dependent problem with unilateral boundary conditions which describes the quasistatic evolution of a rod–spring system under the action of given applied force.

我们考虑了自反性Banach空间中具有单边约束的历史相关变分半融合不等式。不平等的独特可解性来自Sofonea和Migórski（2016，2018）的存在和唯一性结果。在本文中，我们介绍并研究了与不等式相关的广义罚分方法。为此，我们考虑由参数控制的一系列广义罚分问题${\lambda }_{ñ}$ 和一个运算符 $P_{ñ}$。我们证明了惩罚问题的独特可解性以及相应解序列与原始问题解的收敛性。这些结果扩展了Sofonea等人先前的结果。（2018）和Xiao and Sofonea（2019）。最后，我们在单边边界条件的历史相关问题的研究中对它们​​进行了说明，该问题描述了杆-弹簧系统在给定作用力作用下的准静态演化。