ABSTRACT

We consider a semilinear differential variational inequality $\mathcal{P}$ in reflexive Banach spaces, governed by a set of constraints K. We associate to $\mathcal{P}$ a sequence of problems $\left\{{\mathcal{P}}_n\right\}$ where, for each $n\in \mathbb{N}$, ${\mathcal{P}}_n$ is a differential variational inequality governed by a set of constraints $K_n$ and a penalty parameter ${\rho }_n$. We use a result in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1–16. doi:10.1080/00036811.2019.1652736] to prove the unique solvability of problems $\left\{\mathcal{P}\right\}$ and $\left\{{\mathcal{P}}_n\right\}$. Then, we prove that, under appropriate assumptions, the sequence of solutions to Problem ${\mathcal{P}}_n$ converges to the solution of the original problem $\mathcal{P}$. The proof is based on arguments of compactness, pseudomonotonicity and Mosco convergence. We also present two relevant particular case of our convergence result, including a recent result obtained in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1–16. doi:10.1080/00036811.2019.1652736], in the case $K_n=V$. Finally, we provide an example of initial and boundary value problem for which our abstract results can be applied.

摘要

我们考虑半线性微分变分不等式$\mathcal{磷}$在自反巴拿赫空间中，由一组约束K控制。我们联想到$\mathcal{磷}$一系列问题$\left\{{\mathcal{磷}}_n\right\}$其中，对于每个$n\in \mathbb{ñ}$,${\mathcal{磷}}_n$是由一组约束控制的微分变分不等式${ķ}_n$和一个惩罚参数${\rho }_n$. 我们在 [Liu ZH, Zeng SD. 一类微分变分不等式的惩罚方法。应用肛门。2019；1-16。doi:10.1080/00036811.2019.1652736] 证明问题的唯一可解性$\left\{\mathcal{磷}\right\}$和$\left\{{\mathcal{磷}}_n\right\}$. 然后，我们证明，在适当的假设下，问题的解决方案序列${\mathcal{磷}}_n$收敛到原问题的解$\mathcal{磷}$. 该证明基于紧致性、伪单调性和 Mosco 收敛性的论证。我们还介绍了我们收敛结果的两个相关特例，包括最近在 [Liu ZH, Zeng SD. 一类微分变分不等式的惩罚方法。应用肛门。2019；1-16。doi:10.1080/00036811.2019.1652736]，在这种情况下${ķ}_n=五$. 最后，我们提供了一个可以应用我们的抽象结果的初始值和边值问题的示例。