Stability analysis for evolutionary variational-hemivariational inequalities with constraint sets. (English) Zbl 1519.47067
Summary: In this paper, we provide the stability analysis for an evolutionary variational-hemivariational inequality in reflexive Banach space, whose data including the constraint set are perturbed. First, by using its perturbed data and the duality mapping, the perturbed and regularized problems for the evolutionary variational-hemivariational inequality are constructed, respectively. Then, by proving the unique solvability for the evolutionary variational-hemivariational inequality and its perturbed and regularized problems, we obtain two sequences called approximating sequences of the solution to the evolutionary variational-hemivariational inequality, and prove their strong convergence to the unique solution to the evolutionary variational-hemivariational inequality under different mild conditions.
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
| 49J53 | Set-valued and variational analysis |
| 49M29 | Numerical methods involving duality |
Keywords:
evolutionary variational-hemivariational inequality; \(L\)-pseudomonotone; duality mapping; Mosco convergence; smallness condition; strong convergenceReferences:
| [1] | Alleche, B.; Radulescu, V. D.; Sebaoui, M., The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities, Optim Lett, 9, 483-503 (2015) · Zbl 1332.90295 · doi:10.1007/s11590-014-0765-3 |
| [2] | Browder, F. E.; Peter, H., Nonlinear mappings of monotone type in Banach spaces, J Funct Anal, 11, 251-294 (1972) · Zbl 0249.47044 · doi:10.1016/0022-1236(72)90070-5 |
| [3] | Cai, D-L; Sofonea, M.; Xiao, Y-B, Convergence results for elliptic variational-hemivariational inequalities, Adv Nonlinear Anal, 10, 2-23 (2021) · Zbl 1506.47097 · doi:10.1515/anona-2020-0107 |
| [4] | Clarke, F. H., Optimization and Nonsmooth Analysis (1990), Philadelphia: SIAM, Philadelphia · Zbl 0696.49002 · doi:10.1137/1.9781611971309 |
| [5] | Denkowski, Z.; Migórski, S.; Papageorgiou, N. S., An Introduction to Nonlinear Analysis: Theory (2003), Boston: Springer, Boston · Zbl 1040.46001 · doi:10.1007/978-1-4419-9158-4 |
| [6] | Feng, F.; Han, W. M.; Huang, J. G., Virtual element method for an elliptic hemivariational inequality with applications to contact mechanics, J Sci Comput, 81, 2388-2412 (2019) · Zbl 1432.74202 · doi:10.1007/s10915-019-01090-2 |
| [7] | Gasinski, L.; Migórski, S.; Ochal, A., Existence results for evolutionary inclusions and variational-hemivariational inequalities, Appl Anal, 94, 1670-1694 (2015) · Zbl 1317.35135 · doi:10.1080/00036811.2014.940920 |
| [8] | Giannessi, F.; Khan, A. A., Regularization of non-coercive quasi variational inequalities, Control Cybernet, 29, 91-110 (2000) · Zbl 1006.49004 |
| [9] | Han, D. F.; Han, W. M., Numerical analysis of an evolutionary variational-hemivariational inequality with application to a dynamic contact problem, J Comput Appl Math, 358, 163-178 (2019) · Zbl 1460.74070 · doi:10.1016/j.cam.2019.03.010 |
| [10] | Khan, A. A.; Tammer, C.; Zalinescu, C., Regularization of quasi-variational inequalities, Optimization, 64, 1703-1724 (2015) · Zbl 1337.49005 · doi:10.1080/02331934.2015.1028935 |
| [11] | Kulig, A.; Migórski, S., Solvability and continuous dependence results for second order nonlinear evolution inclusions with a Volterra-type operator, Nonlinear Anal, 75, 4729-4746 (2012) · Zbl 1242.35179 · doi:10.1016/j.na.2012.03.023 |
| [12] | Liu, Z. H., Some convergence results for evolution hemivariational inequalities, J Global Optim, 29, 85-95 (2004) · Zbl 1061.49011 · doi:10.1023/B:JOGO.0000035017.75703.7c |
| [13] | Migórski, S.; Dudek, S., Evolutionary Oseen model for generalized Newtonian fluid with multivalued nonmonotone friction law, J Math Fluid Mech, 20, 1317-1333 (2018) · Zbl 06955609 · doi:10.1007/s00021-018-0367-4 |
| [14] | Migoórski, S.; Gamorski, P., A new class of quasistatic frictional contact problems governed by a variational-hemivariational inequality, Nonlinear Anal Real World Appl, 50, 583-602 (2019) · Zbl 1431.49008 · doi:10.1016/j.nonrwa.2019.05.014 |
| [15] | Migórski, S.; Ochal, A.; Sofonea, M., Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems (2013), Berlin: Springer, Berlin · Zbl 1262.49001 · doi:10.1007/978-1-4614-4232-5 |
| [16] | Migórski, S.; Ochal, A.; Sofonea, M., A class of variational-hemivariational inequalities in reflexive Banach spaces, J Elasticity, 127, 151-178 (2017) · Zbl 1368.47045 · doi:10.1007/s10659-016-9600-7 |
| [17] | Naniewicz, Z.; Panagiotopoulos, P. D., Mathematical Theory of Hemivariational Inequalities and Applications (1995), New York: Marcel Dekker, New York · Zbl 0968.49008 |
| [18] | Panagiotopoulos, P. D., Nonconvex energy functions. Hemivariational inequalities and substationarity principles, Acta Mech, 48, 113-130 (1983) · Zbl 0538.73018 · doi:10.1007/BF01170410 |
| [19] | Panagiotopoulos, P. D., Hemivariational Inequalities, Applications in Mechanics and Engineering (1993), Berlin: Springer-Verlag, Berlin · Zbl 0826.73002 |
| [20] | Sofonea, M., Convergence results and optimal control for a class of hemivariational inequalities, SIAM J Math Anal, 50, 4066-4086 (2018) · Zbl 1490.49019 · doi:10.1137/17M1144404 |
| [21] | Sofonea, M.; Migórski, S., Variational-Hemivariational Inequalities with Applications (2018), London: Chapman and Hall/CRC, London · Zbl 1384.49002 |
| [22] | Sofonea, M.; Xiao, Y-B, Tykhonov well-posedness of a viscoplastic contact problem, Evol Equ Control Theory, 9, 1167-1185 (2020) · Zbl 1452.74086 · doi:10.3934/eect.2020048 |
| [23] | Sofonea, M.; Xiao, Y-B; Couderc, M., Optimization problems for a viscoelastic frictional contact problem with unilateral constraints, Nonlinear Anal Real World Appl, 50, 86-103 (2019) · Zbl 1457.74154 · doi:10.1016/j.nonrwa.2019.04.005 |
| [24] | Sofonea, M.; Xiao, Y-B; Couderc, M., Optimization problems for elastic contact models with unilateral constraints, Z Angew Math Phys, 70, 1 (2019) · Zbl 1405.49009 · doi:10.1007/s00033-018-1046-2 |
| [25] | Xiao, Y-B; Huang, N-J, Browder-Tikhonov regularization for a class of evolution second order hemivariational inequalities, J Global Optim, 45, 371-388 (2009) · Zbl 1207.49012 · doi:10.1007/s10898-008-9380-0 |
| [26] | Xiao, Y-B; Sofonea, M., Convergence of solutions to history-dependent variational-hemivariational inequalities, ZAMM Z Angew Math Mech, 99, e201800292 (2019) · Zbl 1538.47100 · doi:10.1002/zamm.201800292 |
| [27] | Xiao, Y-B; Sofonea, M., On the optimal control of variational-hemivariational inequalities, J Math Anal Appl, 475, 364-384 (2019) · Zbl 1414.49020 · doi:10.1016/j.jmaa.2019.02.046 |
| [28] | Xiao, Y-B; Sofonea, M., Generalized penalty method for elliptic variational-hemivariational inequalities, Appl Math Optim, 83, 789-812 (2021) · Zbl 1461.49013 · doi:10.1007/s00245-019-09563-4 |
| [29] | Zeidler, E., Nonlinear Functional Analysis and Its Applications, Volume II (1990), Berlin: Springer-Verlag, Berlin · Zbl 0684.47029 |
| [30] | Zeng, B.; Liu, Z. H.; Migórski, S., On convergence of solutions to variational-hemivariational inequalities, Z Angew Math Phys, 69, 87 (2018) · Zbl 06908352 · doi:10.1007/s00033-018-0980-3 |
| [31] | Zeng, S. D.; Liu, Z. H.; Migórski, S., A class of fractional differential hemivariational inequalities with application to contact problem, Z Angew Math Phys, 69, 36 (2018) · Zbl 1516.35268 · doi:10.1007/s00033-018-0929-6 |
| [32] | Zgurovsky, M. Z.; Kasyanov, P. P.; Zadoianchuk, N-V, Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem, Appl Math Lett, 25, 1569-1574 (2012) · Zbl 1250.49015 · doi:10.1016/j.aml.2012.01.016 |
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