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  December 2020 Early access  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 and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058
References:
[1]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[3]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.

[4]

E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751.  doi: 10.1137/0512062.

[5]

G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

[6]

W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.

[7]

W. Han and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. Numer. Anal., 38 (2000), 556-579.  doi: 10.1137/S0036142998347309.

[8]

P. Kalita, Regularity and Rothe method error estimates for parabolic hemivariational inequality, J. Math. Anal. Appl., 389 (2012), 618-631.  doi: 10.1016/j.jmaa.2011.12.007.

[9]

N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.

[10]

A. Kulig and S. Migóski, Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator, Nonlinear Anal., 75 (2012), 4729-4746.  doi: 10.1016/j.na.2012.03.023.

[11]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[12]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435.  doi: 10.1137/080733231.

[13]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0.

[14]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities: Models and analysis of contact problems, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[15]

S. Migórski and S. Zeng, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.

[16]

S. Migórski and S. Zeng, Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhensive contact model, Nonlinear Anal. Real World Appl., 43 (2018), 121-143.  doi: 10.1016/j.nonrwa.2018.02.008.

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995.

[18]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1.

[19]

Z. Peng, Existence of a class of variational inequalities modelling quasi-static viscoelastic contact problems, ZAMM Z. Angew. Math. Mech., 99 (2019), 20 pp. doi: 10.1002/zamm.201800172.

[20]

Z. Peng, Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation, Nonlinear Anal., 115 (2015), 71-88.  doi: 10.1016/j.na.2014.12.010.

[21]

Z. Peng and Z. Liu, Evolution hemivariational inequality problems with doubly nonlinear operators, J. Global Optim., 51 (2011), 413-427.  doi: 10.1007/s10898-010-9634-5.

[22]

Z. PengZ. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann., 356 (2013), 1339-1358.  doi: 10.1007/s00208-012-0884-z.

[23]

M. RochdiM. Shillor and M. Sofonea, A quasistatic contact problem with directional friction and damped response, Appl. Anal., 68 (1998), 409-422.  doi: 10.1080/00036819808840639.

[24]

M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004.

[25]

M. Sofonea and A. Matei, Mathematical models in contact mechanics, in London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge, 2012.

[26]

M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018.

[27]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

[28]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

show all references

References:
[1]

S. Carl, V. K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.

[3]

Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.

[4]

E. DiBenedetto and R. E. Showalter, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal., 12 (1981), 731-751.  doi: 10.1137/0512062.

[5]

G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, in Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976.

[6]

W. Han and M. Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, in AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.

[7]

W. Han and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. Numer. Anal., 38 (2000), 556-579.  doi: 10.1137/S0036142998347309.

[8]

P. Kalita, Regularity and Rothe method error estimates for parabolic hemivariational inequality, J. Math. Anal. Appl., 389 (2012), 618-631.  doi: 10.1016/j.jmaa.2011.12.007.

[9]

N. Kikuchi and J. T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods, in SIAM Studies in Applied Mathematics, 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.

[10]

A. Kulig and S. Migóski, Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator, Nonlinear Anal., 75 (2012), 4729-4746.  doi: 10.1016/j.na.2012.03.023.

[11]

S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.

[12]

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal., 41 (2009), 1415-1435.  doi: 10.1137/080733231.

[13]

S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0.

[14]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear inclusions and hemivariational inequalities: Models and analysis of contact problems, in Advances in Mechanics and Mathematics, 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[15]

S. Migórski and S. Zeng, Noncoercive hyperbolic variational inequalities with applications to contact mechanics, J. Math. Anal. Appl., 455 (2017), 619-637.  doi: 10.1016/j.jmaa.2017.05.072.

[16]

S. Migórski and S. Zeng, Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhensive contact model, Nonlinear Anal. Real World Appl., 43 (2018), 121-143.  doi: 10.1016/j.nonrwa.2018.02.008.

[17]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, in Monographs and Textbooks in Pure and Applied Mathematics, 188, Marcel Dekker, Inc., New York, 1995.

[18]

P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1.

[19]

Z. Peng, Existence of a class of variational inequalities modelling quasi-static viscoelastic contact problems, ZAMM Z. Angew. Math. Mech., 99 (2019), 20 pp. doi: 10.1002/zamm.201800172.

[20]

Z. Peng, Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation, Nonlinear Anal., 115 (2015), 71-88.  doi: 10.1016/j.na.2014.12.010.

[21]

Z. Peng and Z. Liu, Evolution hemivariational inequality problems with doubly nonlinear operators, J. Global Optim., 51 (2011), 413-427.  doi: 10.1007/s10898-010-9634-5.

[22]

Z. PengZ. Liu and X. Liu, Boundary hemivariational inequality problems with doubly nonlinear operators, Math. Ann., 356 (2013), 1339-1358.  doi: 10.1007/s00208-012-0884-z.

[23]

M. RochdiM. Shillor and M. Sofonea, A quasistatic contact problem with directional friction and damped response, Appl. Anal., 68 (1998), 409-422.  doi: 10.1080/00036819808840639.

[24]

M. Shillor, M. Sofonea and J. J. Telega, Models and analysis of quasistatic contact, in Lecture Notes in Physics, 655, Springer, Berlin, Heidelberg, 2004.

[25]

M. Sofonea and A. Matei, Mathematical models in contact mechanics, in London Mathematical Society Lecture Note Series, 398, Cambridge University Press, Cambridge, 2012.

[26]

M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018.

[27]

E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear Monotone Operators, Springer-verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

[28]

E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

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