We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in[1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.
Liang LU
,
Lijie LI
,
Mircea SOFONEA
. A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES[J]. Acta mathematica scientia, Series B, 2022
, 42(1)
: 247
-264
.
DOI: 10.1007/s10473-022-0114-z
[1] Li X W, Liu Z H, Sofonea M. Unique solvability and exponential stability of differential hemivariational inequalities. Appl Anal, 2020, 99(14):2489-2506
[2] Aubin J P, Cellina A. Differential Inclusions. New York:Springer-Verlag, 1984
[3] Pang J S, Stewart D E. Differential variational inequalities. Math Program, 2008, 113(2):345-424
[4] Chen X J, Wang Z Y. Differential variational inequality approach to dynamic games with shared constraints. Math Program, 2014, 146:379-408
[5] Ke T D, Loi N V, Obukhovskii V. Decay solutions for a class of fractional differential variational inequalities. Fract Calc Appl Anal, 2015, 18(3):531-553
[6] Loi N V. On two parameter global bifurcation of periodic solutions to a class of differential variational inequalities. Nonlinear Anal, 2015, 122:83-99
[7] Loi N V, Ke T D, Obukhovskii V, et al. Topological methods for some classes of differential variational inequalities. J Nonlinear Conv Anal, 2016, 17(3):403-419
[8] Lu L, Liu Z H, Obukhovskii V. Second order differential variational inequalities involving anti-periodic boundary value conditions. J Math Anal Appl, 2019, 473(2):846-865
[9] Gwinner J. On differential variational inequalities and projected dynamical systems-equivalence and a stability result. Discrete Cont Dyn Syst, 2007, 2007(Special):467-476
[10] Gwinner J. On a new class of differential variational inequalities and a stability result. Math Program, 2013, 139(1):205-221
[11] Liu Z H, Migóski S, Zeng S D. Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces. J Differential Equations, 2017, 263(7):3989-4006
[12] Liu Z H, Zeng S D. Differential variational inequalities in infinite Banach spaces. Acta Math Sci, 2017, 37B(1):26-32
[13] Liu Z H, Sofonea M. Differential quasivariational inequalities in contact mechanics. Math Mech Solids, 2019, 24(3):845-861
[14] Liu Z H, Zeng S D, Motreanu D. Evolutionary problems driven by variational inequalities. J Differential Equations, 2016, 260(9):6787-6799
[15] Lu L, Liu Z H, Motreanu D. Existence results of semilinear differential variational inequalities without compactness. Optimization, 2019, 68(5):1017-1035
[16] Nguyen T V, Tran D K. On the differential variational inequalities of parabolic elliptic type. Math Meth Appl Sci, 2017, 40(13):4683-4695
[17] Liu Z H, Zeng S D, Motreanu D. Partial differential hemivariational inequalities. Adv Nonlinear Anal, 2018, 7(4):571-586
[18] Glowinski R. Numerical Methods for Nonlinear Variational Problems. New York:Springer-Verlag, 1984
[19] Sofonea M, Migórski S, Han W. A penalty method for history-dependent variational-hemivariational inequalities. Comput Math Appl, 2018, 75(7):2561-2573
[20] Gwinner J, Jadamba B, Khan A A, et al. Identification in variational and quasi-variational inequalities. J Convex Anal, 2018, 25(2):545-569
[21] Migórski S, Liu Z H, Zeng S D. A class of history-dependent differential variational inequalities with application to contact problems. Optimization, 2020, 69(4):743-775
[22] Liu Z H, Zeng S D. Penalty method for a class of differential variational inequalities. Appl Anal, 2021, 100(7):1574-1589
[23] Pazy A. Semigroups of linear operators and applications to partial differential equations. New York:Springer-Verlag, 1983
[24] Migórski S, Ochal A, Sofonea M. Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26. New York:Springer, 2013
[25] Pascali D, Sburlan S. Nonlinear Mappings of Monotone Type. Netherlands:Springer, 1978
[26] Mosco U. Convergence of convex sets and of solutions of variational inequalities. Adv Math, 1969, 3(4):510-585
[27] Sofonea M, Migórski S. Variational-Hemivariational Inequalities with Applications, Pure and Applied Mathematics. Boca Raton-London:Chapman & Hall/CRC Press, 2018
[28] Clarke F H. Optimization and Nonsmooth Analysis. New York:Wiley, 1983
