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December  2018, 23(10): 4477-4498. doi: 10.3934/dcdsb.2018172

## Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities

 1 College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, Sichuan Province, China 2 Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Chair of Optimization and Control, ul. Lojasiewicza 6, 30-348 Krakow, Poland

* Corresponding author: biao.zeng@outlook.com

Received  May 2017 Revised  January 2018 Published  December 2018 Early access  June 2018

Fund Project: The research is supported by the National Science Center of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, Special Funds of Guangxi Distinguished Experts Construction Engineering, Guangxi, China, and the International Project cofinanced by the Ministry of Science and Higher Education of Republic of Poland under Grant No. 3792/GGPJ/H2020/2017/0.

The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity.

Citation: Stanisław Migórski, Biao Zeng. Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4477-4498. doi: 10.3934/dcdsb.2018172
##### References:
 [1] B. Barabasz, S. Migórski, R. Schaefer and M. Paszynski, Multi deme, twin adaptive strategy $hp$-HGS, Inverse Problems in Science and Engineering, 19 (2011), 3-16.  doi: 10.1080/17415977.2010.531477. [2] B. Barabasz, E. Gajda-Zagorska, S. Migórski, M. Paszynski, R. Schaefer and M. Smolka, A hybrid algorithm for solving inverse problems in elasticity, International Journal of Applied Mathematics and Computer Science, 24 (2014), 865-886.  doi: 10.2478/amcs-2014-0064. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [4] 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. [5] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [6] M. S. Gockenbach and A. A. Khan, An abstract framework for elliptic inverse problems. Ⅰ. an output least-squares approach, Math. Mech. Solids, 12 (2007), 259-276.  doi: 10.1177/1081286505055758. [7] J. Gwinner, B. Jadamba, A. A. Khan and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex Analysis, 25 (2018), 1-25. [8] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal of Mathematical Analysis, 46 (2014), 3891-3912.  doi: 10.1137/140963248. [9] A. Hasanov, Inverse coefficient problems for potential operators, Inverse Problems, 13 (1997), 1265-1278.  doi: 10.1088/0266-5611/13/5/011. [10] M. Hintermüller, Inverse coefficient problems for variational inequalities: Optimality conditions and numerical realization, M2AN Math. Model. Numer. Anal., 35 (2001), 129-152.  doi: 10.1051/m2an:2001109. [11] B. Jadamba, A. A. Khan and M. Sama, Inverse problems of parameter identification in partial differential equations, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011,228-258. doi: 10.1142/9789814338820_0009. [12] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proceedings of the American Mathematical Society, 139 (2011), 1645-1658.  doi: 10.1090/S0002-9939-2010-10594-4. [13] Z. H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim., 72 (2015), 305-323.  doi: 10.1007/s00245-014-9281-1. [14] S. Manservisi and M. Gunzburger, A variational inequality formulation of an inverse elasticity problem, Applied Numerical Mathematics, 34 (2000), 99-126.  doi: 10.1016/S0168-9274(99)00042-2. [15] S. Migórski, Identification of nonlinear heat transfer laws in problems modeled by hemivariational inequalities, in Inverse Problems in Engineering Mechanics, (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 1998, 27-36. doi: 10.1016/B978-008043319-6/50007-8. [16] S. Migórski, Sensitivity analysis of inverse problems with applications to nonlinear systems, Dynamic Systems and Applications, 8 (1999), 73-88. [17] S. Migórski, Identification coefficient problems for elliptic hemivariational inequalities and applications, in Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 2000. [18] S. Migórski, Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Problems in Engineering, 11 (2003), 229-242. [19] S. Migórski, Identification of operators in systems governed by second order evolution inclusions with applications to hemivariational inequalities, International Journal of Innovative Computing, Information and Control, 8 (2012), 3845-3862. [20] S. Migórski and A. Ochal, Inverse coefficient problem for elliptic hemivariational inequality, in Nonsmooth/Nonconvex Mechanics, Modeling, Analysis and Numerical Methods (eds. D. Y. Gao et al.), Kluwer Academic Publishers, 50 (2001), 247-261. doi: 10.1007/978-1-4613-0275-9_11. [21] S. Migórski and A. Ochal, An inverse coefficient problem for a parabolic hemivariational inequality, Applicable Analysis, 89 (2010), 243-256.  doi: 10.1080/00036810902889559. [22] 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. [23] S. Migórski, A. Ochal and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity, 127 (2017), 151-178.  doi: 10.1007/s10659-016-9600-7. [24] S. Migórski and B. Zeng, Variational-hemivariational inverse problems for unilateral frictional contact, Applicable Analysis, (2018). doi: 10.1080/00036811.2018.1491037. [25] D. Motreanu and M. Sofonea, Quasivariational inequalities and applications in frictional contact problems with normal compliance, Adv. Math. Sci. Appl., 10 (2000), 103-118. [26] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. [27] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [28] M. Sofonea, W. Han and S. Migórski, Numerical analysis of history-dependent variational inequalities with applications to contact problems, European Journal of Applied Mathematics, 26 (2015), 427-452.  doi: 10.1017/S095679251500011X. [29] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398 Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. [30] M. Sofonea, S. Migórski, Variational-Hemivariational Inequalities with Applications, Chapman & Hall/CRC, Monographs and Research Notes in Mathematics, Boca Raton, 2017. [31] M. Sofonea and F. Patrulescu, Penalization of history-dependent variational inequalities, European Journal of Applied Mathematics, 25 (2014), 155-176.  doi: 10.1017/S0956792513000363. [32] E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

##### References:
 [1] B. Barabasz, S. Migórski, R. Schaefer and M. Paszynski, Multi deme, twin adaptive strategy $hp$-HGS, Inverse Problems in Science and Engineering, 19 (2011), 3-16.  doi: 10.1080/17415977.2010.531477. [2] B. Barabasz, E. Gajda-Zagorska, S. Migórski, M. Paszynski, R. Schaefer and M. Smolka, A hybrid algorithm for solving inverse problems in elasticity, International Journal of Applied Mathematics and Computer Science, 24 (2014), 865-886.  doi: 10.2478/amcs-2014-0064. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. [4] 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. [5] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003. [6] M. S. Gockenbach and A. A. Khan, An abstract framework for elliptic inverse problems. Ⅰ. an output least-squares approach, Math. Mech. Solids, 12 (2007), 259-276.  doi: 10.1177/1081286505055758. [7] J. Gwinner, B. Jadamba, A. A. Khan and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex Analysis, 25 (2018), 1-25. [8] W. Han, S. Migórski and M. Sofonea, A class of variational-hemivariational inequalities with applications to frictional contact problems, SIAM Journal of Mathematical Analysis, 46 (2014), 3891-3912.  doi: 10.1137/140963248. [9] A. Hasanov, Inverse coefficient problems for potential operators, Inverse Problems, 13 (1997), 1265-1278.  doi: 10.1088/0266-5611/13/5/011. [10] M. Hintermüller, Inverse coefficient problems for variational inequalities: Optimality conditions and numerical realization, M2AN Math. Model. Numer. Anal., 35 (2001), 129-152.  doi: 10.1051/m2an:2001109. [11] B. Jadamba, A. A. Khan and M. Sama, Inverse problems of parameter identification in partial differential equations, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011,228-258. doi: 10.1142/9789814338820_0009. [12] V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proceedings of the American Mathematical Society, 139 (2011), 1645-1658.  doi: 10.1090/S0002-9939-2010-10594-4. [13] Z. H. Liu and B. Zeng, Optimal control of generalized quasi-variational hemivariational inequalities and its applications, Appl. Math. Optim., 72 (2015), 305-323.  doi: 10.1007/s00245-014-9281-1. [14] S. Manservisi and M. Gunzburger, A variational inequality formulation of an inverse elasticity problem, Applied Numerical Mathematics, 34 (2000), 99-126.  doi: 10.1016/S0168-9274(99)00042-2. [15] S. Migórski, Identification of nonlinear heat transfer laws in problems modeled by hemivariational inequalities, in Inverse Problems in Engineering Mechanics, (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 1998, 27-36. doi: 10.1016/B978-008043319-6/50007-8. [16] S. Migórski, Sensitivity analysis of inverse problems with applications to nonlinear systems, Dynamic Systems and Applications, 8 (1999), 73-88. [17] S. Migórski, Identification coefficient problems for elliptic hemivariational inequalities and applications, in Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G. S. Dulikravich), Elsevier, 2000. [18] S. Migórski, Homogenization technique in inverse problems for boundary hemivariational inequalities, Inverse Problems in Engineering, 11 (2003), 229-242. [19] S. Migórski, Identification of operators in systems governed by second order evolution inclusions with applications to hemivariational inequalities, International Journal of Innovative Computing, Information and Control, 8 (2012), 3845-3862. [20] S. Migórski and A. Ochal, Inverse coefficient problem for elliptic hemivariational inequality, in Nonsmooth/Nonconvex Mechanics, Modeling, Analysis and Numerical Methods (eds. D. Y. Gao et al.), Kluwer Academic Publishers, 50 (2001), 247-261. doi: 10.1007/978-1-4613-0275-9_11. [21] S. Migórski and A. Ochal, An inverse coefficient problem for a parabolic hemivariational inequality, Applicable Analysis, 89 (2010), 243-256.  doi: 10.1080/00036810902889559. [22] 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. [23] S. Migórski, A. Ochal and M. Sofonea, A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity, 127 (2017), 151-178.  doi: 10.1007/s10659-016-9600-7. [24] S. Migórski and B. Zeng, Variational-hemivariational inverse problems for unilateral frictional contact, Applicable Analysis, (2018). doi: 10.1080/00036811.2018.1491037. [25] D. Motreanu and M. Sofonea, Quasivariational inequalities and applications in frictional contact problems with normal compliance, Adv. Math. Sci. Appl., 10 (2000), 103-118. [26] Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, Basel, Hong Kong, 1995. [27] P. D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [28] M. Sofonea, W. Han and S. Migórski, Numerical analysis of history-dependent variational inequalities with applications to contact problems, European Journal of Applied Mathematics, 26 (2015), 427-452.  doi: 10.1017/S095679251500011X. [29] M. Sofonea and A. Matei, Mathematical Models in Contact Mechanics, London Mathematical Society Lecture Note Series, 398 Cambridge University Press, 2012. doi: 10.1017/CBO9781139104166. [30] M. Sofonea, S. Migórski, Variational-Hemivariational Inequalities with Applications, Chapman & Hall/CRC, Monographs and Research Notes in Mathematics, Boca Raton, 2017. [31] M. Sofonea and F. Patrulescu, Penalization of history-dependent variational inequalities, European Journal of Applied Mathematics, 25 (2014), 155-176.  doi: 10.1017/S0956792513000363. [32] E. Zeidler, Nonlinear Functional Analysis and Applications II A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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