# 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 and Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044
##### References:
 [1] X. J. Chen and Z. Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), Ser. A, 379–408. doi: 10.1007/s10107-013-0689-1. [2] X. J. Chen and Z. Y. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim., 23 (2013), 1647-1671.  doi: 10.1137/120875223. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [5] J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program., 139 (2013), 205-221.  doi: 10.1007/s10107-013-0669-5. [6] K. L. Kuttler and M. Shillor, Dynamic contact with normal compliance wear and discontinuous friction coefficient, SIAM J. Math. Anal., 34 (2002), 1-27.  doi: 10.1137/S0036141001391184. [7] Z. H. Liu and M. Sofonea, Differential quasivariational inequalities in contact mechanics, Math. Mech. Solids, 24 (2019), 845-861.  doi: 10.1177/1081286518755563. [8] Z. H. Liu, D. Motreanu and S. D. Zeng, Nonlinear evolutionary systems driven by mixed variational inequalities and its applications, Nonlinear Anal. Real World Appl., 42 (2018), 409-421.  doi: 10.1016/j.nonrwa.2018.01.008. [9] Z. H. Liu, S. Migórski and S. D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations, 263 (2017), 3989-4006.  doi: 10.1016/j.jde.2017.05.010. [10] Z. H. Liu, D. Motreanu and S. D. Zeng, On the well-posedness of differential mixed quasi-variational inequalities, Topol. Method Nonl. Anal., 51 (2018), 135-150. [11] Z. H. Liu, S. D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012. [12] Z. H. Liu and S. D. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Math. Sci. Ser. B, 37 (2017), 26-32.  doi: 10.1016/S0252-9602(16)30112-6. [13] Z. H. Liu, S. D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal., 7 (2018), 571-586.  doi: 10.1515/anona-2016-0102. [14] N. V. Loi, On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal., 122 (2015), 83-99.  doi: 10.1016/j.na.2015.03.019. [15] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [16] S. Migórski and S. D. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Glob. Optim., 72 (2018), 761-779.  doi: 10.1007/s10898-018-0667-5. [17] S. Migórski and S. D. Zeng, Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model, Nonlinear Anal. Real World Appl., 43 (2018), 121-143.  doi: 10.1016/j.nonrwa.2018.02.008. [18] S. Migórski and S. D. Zeng, A class of generalized evolutionary problems driven by variational inequalities and fractional operators, Set-Valued Var. Anal., 27 (2019), 949–970. https://doi.org/10.1007/s11228-018-0502-7. doi: 10.1007/s11228-018-0502-7. [19] S. Migórski and S. D. Zeng, Mixed variational inequalities driven by fractional evolution equations, ACTA Math. Sci., 39 (2019), 461-468. [20] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), Ser. A, 345–424. doi: 10.1007/s10107-006-0052-x. [21] M. Shillor, M. Sofonea and J. J. Telega, Analysis of viscoelastic contact with normal compliance, friction and wear diffusion, Comptes Rendus Mecanique, 331 (2003), 395-400. [22] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [23] M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Comm. Pure Appl. Anal., 7 (2008), 645-658.  doi: 10.3934/cpaa.2008.7.645. [24] M. Sofonea, F. P\v{a}trulescu and Y. Souleiman, Analysis of a contact problem with wear and unilateral constraint, Appl. Anal., 95 (2016), 2590-2607.  doi: 10.1080/00036811.2015.1102892. [25] N. T. Van Anh and T. D. Ke, Asymptotic behavior of solutions to a class of differential variational inequalities, Ann. Polon. Math., 114 (2015), 147-164.  doi: 10.4064/ap114-2-5. [26] S. D. Zeng, Z. H. Liu and S. Migorski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys., 69 (2018), Art. 36, 23 pp. doi: 10.1007/s00033-018-0929-6.

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##### References:
 [1] X. J. Chen and Z. Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program., 146 (2014), Ser. A, 379–408. doi: 10.1007/s10107-013-0689-1. [2] X. J. Chen and Z. Y. Wang, Convergence of regularized time-stepping methods for differential variational inequalities, SIAM J. Optim., 23 (2013), 1647-1671.  doi: 10.1137/120875223. [3] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. doi: 10.1007/978-1-4419-9158-4. [4] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [5] J. Gwinner, On a new class of differential variational inequalities and a stability result, Math. Program., 139 (2013), 205-221.  doi: 10.1007/s10107-013-0669-5. [6] K. L. Kuttler and M. Shillor, Dynamic contact with normal compliance wear and discontinuous friction coefficient, SIAM J. Math. Anal., 34 (2002), 1-27.  doi: 10.1137/S0036141001391184. [7] Z. H. Liu and M. Sofonea, Differential quasivariational inequalities in contact mechanics, Math. Mech. Solids, 24 (2019), 845-861.  doi: 10.1177/1081286518755563. [8] Z. H. Liu, D. Motreanu and S. D. Zeng, Nonlinear evolutionary systems driven by mixed variational inequalities and its applications, Nonlinear Anal. Real World Appl., 42 (2018), 409-421.  doi: 10.1016/j.nonrwa.2018.01.008. [9] Z. H. Liu, S. Migórski and S. D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations, 263 (2017), 3989-4006.  doi: 10.1016/j.jde.2017.05.010. [10] Z. H. Liu, D. Motreanu and S. D. Zeng, On the well-posedness of differential mixed quasi-variational inequalities, Topol. Method Nonl. Anal., 51 (2018), 135-150. [11] Z. H. Liu, S. D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations, 260 (2016), 6787-6799.  doi: 10.1016/j.jde.2016.01.012. [12] Z. H. Liu and S. D. Zeng, Differential variational inequalities in infinite Banach spaces, Acta Math. Sci. Ser. B, 37 (2017), 26-32.  doi: 10.1016/S0252-9602(16)30112-6. [13] Z. H. Liu, S. D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal., 7 (2018), 571-586.  doi: 10.1515/anona-2016-0102. [14] N. V. Loi, On two-parameter global bifurcation of periodic solutions to a class of differential variational inequalities, Nonlinear Anal., 122 (2015), 83-99.  doi: 10.1016/j.na.2015.03.019. [15] S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5. [16] S. Migórski and S. D. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Glob. Optim., 72 (2018), 761-779.  doi: 10.1007/s10898-018-0667-5. [17] S. Migórski and S. D. Zeng, Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model, Nonlinear Anal. Real World Appl., 43 (2018), 121-143.  doi: 10.1016/j.nonrwa.2018.02.008. [18] S. Migórski and S. D. Zeng, A class of generalized evolutionary problems driven by variational inequalities and fractional operators, Set-Valued Var. Anal., 27 (2019), 949–970. https://doi.org/10.1007/s11228-018-0502-7. doi: 10.1007/s11228-018-0502-7. [19] S. Migórski and S. D. Zeng, Mixed variational inequalities driven by fractional evolution equations, ACTA Math. Sci., 39 (2019), 461-468. [20] J.-S. Pang and D. E. Stewart, Differential variational inequalities, Math. Program., 113 (2008), Ser. A, 345–424. doi: 10.1007/s10107-006-0052-x. [21] M. Shillor, M. Sofonea and J. J. Telega, Analysis of viscoelastic contact with normal compliance, friction and wear diffusion, Comptes Rendus Mecanique, 331 (2003), 395-400. [22] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [23] M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems, Comm. Pure Appl. Anal., 7 (2008), 645-658.  doi: 10.3934/cpaa.2008.7.645. [24] M. Sofonea, F. P\v{a}trulescu and Y. Souleiman, Analysis of a contact problem with wear and unilateral constraint, Appl. Anal., 95 (2016), 2590-2607.  doi: 10.1080/00036811.2015.1102892. [25] N. T. Van Anh and T. D. Ke, Asymptotic behavior of solutions to a class of differential variational inequalities, Ann. Polon. Math., 114 (2015), 147-164.  doi: 10.4064/ap114-2-5. [26] S. D. Zeng, Z. H. Liu and S. Migorski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys., 69 (2018), Art. 36, 23 pp. doi: 10.1007/s00033-018-0929-6.
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