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. LiuD. 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. LiuS. 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. LiuD. 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. LiuS. 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. LiuS. 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. ShillorM. 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. SofoneaC. 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. SofoneaF. 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.

show all references

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. LiuD. 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. LiuS. 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. LiuD. 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. LiuS. 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. LiuS. 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. ShillorM. 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. SofoneaC. 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. SofoneaF. 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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