# American Institute of Mathematical Sciences

December  2020, 9(4): 1153-1165. doi: 10.3934/eect.2020058

## Existence for a quasistatic variational-hemivariational inequality

 Guangxi Key Laboratory of Universities Optimization Control and Engineering, Calculation, and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

* Corresponding author: Zijia Peng

Received  November 2019 Revised  February 2020 Published  May 2020

Fund Project: This work is supported by the NNSF of China grant Nos. 11901122 and 11561007, the NSF of Guangxi, China grant No. 2018GXNSFAA050008, and the Xiangsihu Young Scholars and Innovative Research Team of GXUN, China grant No. 2018RSCXSHQN04. The first author is also supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH

This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migórski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded subdifferential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications.

Citation: Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations & Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
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