# American Institute of Mathematical Sciences

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  December 2020 Early access  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 and 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. [2] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [4] M. M. Čoban, P. S. Kenderov and J. P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika, 36 (1989), 301-324.  doi: 10.1112/S0025579300013152. [5] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes Mathematics, 1543. Springer, Berlin, 1993. doi: 10.1007/BFb0084195. [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. [7] C. Eck, J. 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. [8] Y.-P. Fang, N.-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. [9] D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909-921.  doi: 10.1080/01630569508816652. [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. [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. [12] J. Haslinger, I. 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. [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. [14] R. Hu, Y.-B. Xiao, N.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. [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. [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. [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. [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. [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. [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. [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. [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. [23] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1. [24] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [25] A. Petruşel, I. 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. [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. [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. [28] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398. Cambridge University Press, 2012. [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. [30] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [31] M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electronic Journal of Differential Equations, 2019 (2019), 19 pp. [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. [33] A. N. Tykhonov, On the stability of functional optimization problems, USSR Comput. Math. Math. Phys., 6 (1966), 631-634. [34] Y.-M. Wang, Y.-B. Xiao, X. 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. [35] Y.-B. Xiao, N.-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. [36] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.

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##### 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. [2] F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc., 205 (1975), 247-262.  doi: 10.1090/S0002-9947-1975-0367131-6. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [4] M. M. Čoban, P. S. Kenderov and J. P. Revalski, Generic well-posedness of optimization problems in topological spaces, Mathematika, 36 (1989), 301-324.  doi: 10.1112/S0025579300013152. [5] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes Mathematics, 1543. Springer, Berlin, 1993. doi: 10.1007/BFb0084195. [6] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976. [7] C. Eck, J. 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. [8] Y.-P. Fang, N.-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. [9] D. Goeleven and D. Mentagui, Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim., 16 (1995), 909-921.  doi: 10.1080/01630569508816652. [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. [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. [12] J. Haslinger, I. 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. [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. [14] R. Hu, Y.-B. Xiao, N.-J. Huang and X. Wang, Equivalence results of well-posedness for split variational-hemivariational inequalities, J. Nonlinear Convex Anal., 20 (2019), 447-459. [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. [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. [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. [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. [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. [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. [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. [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. [23] P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications, Birkhäuser Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1. [24] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [25] A. Petruşel, I. 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. [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. [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. [28] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398. Cambridge University Press, 2012. [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. [30] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. [31] M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electronic Journal of Differential Equations, 2019 (2019), 19 pp. [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. [33] A. N. Tykhonov, On the stability of functional optimization problems, USSR Comput. Math. Math. Phys., 6 (1966), 631-634. [34] Y.-M. Wang, Y.-B. Xiao, X. 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. [35] Y.-B. Xiao, N.-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. [36] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.
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