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A generalized penalty method for differential variational-hemivariational inequalities

Abstract

We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in [1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.

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Correspondence to Mircea Sofonea.

Additional information

This work was supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement (823731 CONMECH). The first author is also supported by National Natural Science Foundation of China (11671101), Guangxi Natural Science Foundation (2021GXNSFAA075022) and Project of Guangxi Education Department (2020KY16017). The second author is also supported by National Natural Science Foundation of China (11961074) and Yulin normal university of scientific research fund for high-level talents (G2019ZK39, G2021ZK06).

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Lu, L., Li, L. & Sofonea, M. A generalized penalty method for differential variational-hemivariational inequalities. Acta Math Sci 42, 247–264 (2022). https://doi.org/10.1007/s10473-022-0114-z

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  • DOI: https://doi.org/10.1007/s10473-022-0114-z

Key words

  • differential variational-hemivariational inequality
  • generalized penalty method
  • Mosco convergence
  • viscoelastic rod
  • unilateral constraint

2010 MR Subject Classification

  • 34G20
  • 47J20
  • 49J40
  • 49J45
  • 74K10
  • 74M15