# American Institute of Mathematical Sciences

December  2020, 9(4): 1073-1087. doi: 10.3934/eect.2020044

## History-dependent differential variational-hemivariational inequalities with applications to contact mechanics

 1 College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi, China 2 Guangxi Colleges and Universities Key Laboratory of Complex System Optimization, and Big Data Processing, Yulin Normal University, Yulin 537000, China 3 Departement of Mathematics, FPT University, Education zone, Hoa Lac high tech park, Km29 Thang Long highway, Thach That ward, Hanoi, Vietnam 4 Center for General Education, China Medical University, Taichung, Taiwan 5 Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30348 Krakow, Poland

* Corresponding author: Shengda Zeng

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday.

Received  September 2019 Published  December 2020 Early access  March 2020

Fund Project: This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 – CONMECH. It is also supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611, NNSF of China Grant No. 11671101, NSF of Guangxi Grant No. 2018GXNSFDA138002, and International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0

The primary objective of this paper is to explore a complicated differential variational-hemivariational inequality involving a history-dependent operator in Banach spaces. A well-posedness result for the inequality, including the existence, uniqueness, and continuous dependence on the initial data of the solution is established by using a fixed point principle for history-dependent operators. Moreover, to illustrate the applicability of the theoretical results, an elastic contact problem with wear and long time dependent effort is explored.

Citation: Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044
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##### References:
 [1] 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 [2] Stanisław Migórski, Yi-bin Xiao, Jing Zhao. Fully history-dependent evolution hemivariational inequalities with constraints. Evolution Equations & Control Theory, 2020, 9 (4) : 1089-1114. doi: 10.3934/eect.2020047 [3] Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure & Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371 [4] Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036 [5] Rong Hu, Ya-Ping Fang, Nan-Jing Huang. Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints. Journal of Industrial & Management Optimization, 2010, 6 (3) : 465-481. doi: 10.3934/jimo.2010.6.465 [6] Jian-Wen Peng, Xin-Min Yang. Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (3) : 701-714. doi: 10.3934/jimo.2015.11.701 [7] X. X. Huang, Xiaoqi Yang. Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2007, 3 (4) : 671-684. doi: 10.3934/jimo.2007.3.671 [8] Xiaoliang Cheng, Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of two quasistatic history-dependent contact models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2425-2445. doi: 10.3934/dcdsb.2014.19.2425 [9] Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172 [10] Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks & Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601 [11] Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813 [12] Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026 [13] Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567 [14] Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819 [15] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [16] Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112 [17] Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737 [18] Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021225 [19] Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292 [20] Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259

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