December  2020, 9(4): 1167-1185. doi: 10.3934/eect.2020048

Tykhonov well-posedness of a viscoplastic contact problem

1. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731, China

2. 

Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France

* Corresponding author: Mircea Sofonea

†This paper is dedicated to Professor Meir Shillor on the occasion of his 70th birthday.

Received  October 2019 Published  March 2020

We consider an initial and boundary value problem $ {{\mathcal{P}}} $ which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is $ \mathbb{R}_+ = [0,+\infty) $. We list the assumptions on the data and derive a variational formulation ${{\mathcal{P}}}_V $ of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple $ {{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}}) $ where $ I $ is set of parameters, $ \Omega $ represents a family of approximating sets and ${{\mathcal{C} }} $ is a set of sequences, then we define the well-posedness of Problem ${{\mathcal{P}}}_V $ with respect to $ {{\mathcal{T}}}$. Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of $ {{\mathcal{P} }}_V $ with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem $ {{\mathcal{P}}} $ can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.

Citation: Mircea Sofonea, Yi-bin Xiao. Tykhonov well-posedness of a viscoplastic contact problem. Evolution Equations & Control Theory, 2020, 9 (4) : 1167-1185. doi: 10.3934/eect.2020048
References:
[1]

A. Capatina, Variational Inequalities Frictional Contact Problems, Advances in Mechanics and Mathematics, 31. Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

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F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

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M. M. ČobanP. S. Kenderov and J. P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika, 36 (1989), 301-324.  doi: 10.1112/S0025579300013152.  Google Scholar

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A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes Mathematics, 1543. Springer, Berlin, 1993. doi: 10.1007/BFb0084195.  Google Scholar

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G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[7] C. EckJ. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, 270. Chapman/CRC Press, New York, 2005.  doi: 10.1201/9781420027365.  Google Scholar
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Y.-P. FangN.-J. Huang and J.-C. Yao, Well-posedness by perturbations of mixed variational inequalities in Banach spaces, Eur. J. Oper. Res., 201 (2010), 682-692.  doi: 10.1016/j.ejor.2009.04.001.  Google Scholar

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D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909-921.  doi: 10.1080/01630569508816652.  Google Scholar

[10] W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002.   Google Scholar
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W. M. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numer., 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar

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J. HaslingerI. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485.   Google Scholar

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I. Hlaváček, J. Haslinger, J. Necǎs and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.  Google Scholar

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R. HuY.-B. XiaoN.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459.   Google Scholar

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X. X. Huang, Extended and strongly extended well-posedness of set-valued optimization problems, Math. Methods Oper. Res., 53 (2001), 101-116.  doi: 10.1007/s001860000100.  Google Scholar

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X. X. Huang and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.  doi: 10.1137/040614943.  Google Scholar

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N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

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R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.  doi: 10.1080/01630568108816100.  Google Scholar

[19]

R. Lucchetti and F. Patrone, Some properties of "wellposedness" variational inequalities governed by linear operators, Numer. Funct. Anal. Optim., 5 (1982/83), 349-361.  doi: 10.1080/01630568308816145.  Google Scholar

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R. Lucchetti, Convexity and Well-Posed Problems, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 22. Springer, New York, 2006. doi: 10.1007/0-387-31082-7.  Google Scholar

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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.  Google Scholar

[22]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.  Google Scholar

[23]

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

[24]

P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar

[25]

A. PetruşelI. A. Rus and J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), 903-914.  doi: 10.11650/twjm/1500404764.  Google Scholar

[26]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes Physics, 655. Springer, Berlin, 2004. doi: 10.1007/b99799.  Google Scholar

[27]

Q.-Y. Shu, R. Hu and Y.-B. Xiao, Metric characterizations for well-posedness of split hemivariational inequalities, Journal of Inequalities and Applications, 2018 (2018), 17 pp. doi: 10.1186/s13660-018-1761-4.  Google Scholar

[28] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398. Cambridge University Press, 2012.   Google Scholar
[29]

M. Sofonea, A. Matei and Y.-B. Xiao, Optimal control for a class of mixed variational problems, Z. Angew. Math. Phys., 70 (2019), Art. 127, 17 pp. doi: 10.1007/s00033-019-1173-4.  Google Scholar

[30] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018.   Google Scholar
[31]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electronic Journal of Differential Equations, 2019 (2019), 19 pp.  Google Scholar

[32]

M. Sofonea and Y.-B. Xiao, On the well-posedness concept in the sense of Tykhonov, J. Optim. Theory Appl., 183 (2019), 139-157.  doi: 10.1007/s10957-019-01549-0.  Google Scholar

[33]

A. N. Tykhonov, On the stability of functional optimization problems, USSR Comput. Math. Math. Phys., 6 (1966), 631-634.   Google Scholar

[34]

Y.-M. WangY.-B. XiaoX. Wang and Y. J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.  Google Scholar

[35]

Y.-B. XiaoN.-J. Huang and M.-M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math., 15 (2011), 1261-1276.  doi: 10.11650/twjm/1500406298.  Google Scholar

[36]

T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.  Google Scholar

show all references

References:
[1]

A. Capatina, Variational Inequalities Frictional Contact Problems, Advances in Mechanics and Mathematics, 31. Springer, New York, 2014. doi: 10.1007/978-3-319-10163-7.  Google Scholar

[2]

F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6.  Google Scholar

[3]

F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.  Google Scholar

[4]

M. M. ČobanP. S. Kenderov and J. P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika, 36 (1989), 301-324.  doi: 10.1112/S0025579300013152.  Google Scholar

[5]

A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes Mathematics, 1543. Springer, Berlin, 1993. doi: 10.1007/BFb0084195.  Google Scholar

[6]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[7] C. EckJ. Jarušek and M. Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, 270. Chapman/CRC Press, New York, 2005.  doi: 10.1201/9781420027365.  Google Scholar
[8]

Y.-P. FangN.-J. Huang and J.-C. Yao, Well-posedness by perturbations of mixed variational inequalities in Banach spaces, Eur. J. Oper. Res., 201 (2010), 682-692.  doi: 10.1016/j.ejor.2009.04.001.  Google Scholar

[9]

D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909-921.  doi: 10.1080/01630569508816652.  Google Scholar

[10] W. M. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002.   Google Scholar
[11]

W. M. Han and M. Sofonea, Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numer., 28 (2019), 175-286.  doi: 10.1017/S0962492919000023.  Google Scholar

[12]

J. HaslingerI. Hlaváček and J. Nečas, Numerical methods for unilateral problems in solid mechanics, Handbook of Numerical Analysis, Handb. Numer. Anal., North-Holland, Amsterdam, 4 (1996), 313-485.   Google Scholar

[13]

I. Hlaváček, J. Haslinger, J. Necǎs and J. Lovíšek, Solution of Variational Inequalities in Mechanics, Mathematical Sciences, 66. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1048-1.  Google Scholar

[14]

R. HuY.-B. XiaoN.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459.   Google Scholar

[15]

X. X. Huang, Extended and strongly extended well-posedness of set-valued optimization problems, Math. Methods Oper. Res., 53 (2001), 101-116.  doi: 10.1007/s001860000100.  Google Scholar

[16]

X. X. Huang and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim., 17 (2006), 243-258.  doi: 10.1137/040614943.  Google Scholar

[17]

N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics, 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970845.  Google Scholar

[18]

R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.  doi: 10.1080/01630568108816100.  Google Scholar

[19]

R. Lucchetti and F. Patrone, Some properties of "wellposedness" variational inequalities governed by linear operators, Numer. Funct. Anal. Optim., 5 (1982/83), 349-361.  doi: 10.1080/01630568308816145.  Google Scholar

[20]

R. Lucchetti, Convexity and Well-Posed Problems, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 22. Springer, New York, 2006. doi: 10.1007/0-387-31082-7.  Google Scholar

[21]

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.  Google Scholar

[22]

Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 188. Marcel Dekker, Inc., New York, 1995.  Google Scholar

[23]

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

[24]

P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.  Google Scholar

[25]

A. PetruşelI. A. Rus and J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), 903-914.  doi: 10.11650/twjm/1500404764.  Google Scholar

[26]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Lecture Notes Physics, 655. Springer, Berlin, 2004. doi: 10.1007/b99799.  Google Scholar

[27]

Q.-Y. Shu, R. Hu and Y.-B. Xiao, Metric characterizations for well-posedness of split hemivariational inequalities, Journal of Inequalities and Applications, 2018 (2018), 17 pp. doi: 10.1186/s13660-018-1761-4.  Google Scholar

[28] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398. Cambridge University Press, 2012.   Google Scholar
[29]

M. Sofonea, A. Matei and Y.-B. Xiao, Optimal control for a class of mixed variational problems, Z. Angew. Math. Phys., 70 (2019), Art. 127, 17 pp. doi: 10.1007/s00033-019-1173-4.  Google Scholar

[30] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018.   Google Scholar
[31]

M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electronic Journal of Differential Equations, 2019 (2019), 19 pp.  Google Scholar

[32]

M. Sofonea and Y.-B. Xiao, On the well-posedness concept in the sense of Tykhonov, J. Optim. Theory Appl., 183 (2019), 139-157.  doi: 10.1007/s10957-019-01549-0.  Google Scholar

[33]

A. N. Tykhonov, On the stability of functional optimization problems, USSR Comput. Math. Math. Phys., 6 (1966), 631-634.   Google Scholar

[34]

Y.-M. WangY.-B. XiaoX. Wang and Y. J. Cho, Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems, J. Nonlinear Sci. Appl., 9 (2016), 1178-1192.  doi: 10.22436/jnsa.009.03.44.  Google Scholar

[35]

Y.-B. XiaoN.-J. Huang and M.-M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math., 15 (2011), 1261-1276.  doi: 10.11650/twjm/1500406298.  Google Scholar

[36]

T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.  Google Scholar

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