Generalized penalty method for semilinear differential variational inequalities

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.

Keywords:

• Differential variational inequality
• unilateral constraint
• generalized penalty method
• Mosco convergence
• initial and boundary value problem

2010 Mathematics Subject Classifications:

• 47J20
• 47J35
• 49J30
• 49J40
• 35A15

Acknowledgments

This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH. It is also supported by the National Natural Science Foundation of China under the Grant Nos. 11671101 and 11961074, the Guangxi Natural Science Foundation under the Grant No. 2017GXNSFBA198031, and the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 823731 CONMECH. It is also supported by the National Natural Science Foundation of China [grant numbers 11671101 and 11961074], the Guangxi Natural Science Foundation [grant number 2017GXNSFBA198031], and the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35).