Advanced Search
Article Contents
Article Contents

Fully history-dependent evolution hemivariational inequalities with constraints

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

    Mathematics Subject Classification: Primary: 47J20, 47J22, 49J40, 49J45; Secondary: 74G25, 74G30, 74M15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J. Ahn and D.E. Stewart, Dynamic frictionless contact in linear viscoelasticity, IMA J. Numer. Anal., 29 (2009), 43-71.  doi: 10.1093/imanum/drm029.
    [2] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145. 
    [3] S. Carl, V.K. Le and D. Motreanu, Nonsmooth Variational Problems and their Inequalities. Comparison Principles and Applications, Springer Monographs in Mathematics. Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3.
    [4] S. CarlV.K. Le and D. Motreanu, Evolutionary variational-hemivariational inequalities: Existence and comparison results, J. Math. Anal. Appl., 345 (2008), 545-558.  doi: 10.1016/j.jmaa.2008.04.005.
    [5] O. ChadliQ.H. Ansari and S. Al-Homidan, Existence of solutions for nonlinear implicit differential equations: An equilibrium problem approach, Numer. Func. Anal. Optim., 37 (2016), 1385-1419.  doi: 10.1080/01630563.2016.1210164.
    [6] O. ChadliQ.H. Ansari and J.-C. Yao, Mixed equilibrium problems and anti-periodic solutions for nonlinear evolution equations, J. Optim. Theory Appl., 168 (2016), 410-440.  doi: 10.1007/s10957-015-0707-y.
    [7] F.H. Clarke, Optimization and Nonsmooth Analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.
    [8] M. Cocou, Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity, Z. angew. Math. Phys., 53 (2002), 1099-1109.  doi: 10.1007/PL00012615.
    [9] 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.
    [10] Z. Denkowski, S. Migórski and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic Publishers, Boston, MA, 2003.
    [11] C. EckJ. Jarušek and  M. KrbečUnilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics, 270. Chapman/CRC Press, New York, Boca Raton, FL, 2005.  doi: 10.1201/9781420027365.
    [12] C. EckJ. Jarušek and M. Sofonea, A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction, European J. Appl. Math., 21 (2010), 229-251.  doi: 10.1017/S0956792510000045.
    [13] E.-H. Essoufi and M. Kabbaj, Existence of solutions of a dynamic Signorini's problem with nonlocal friction for viscoelastic piezoelectric materials, Bull. Math. Soc. Sc. Math. Roumanie (N.S.), 48 (2005), 181-195. 
    [14] D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities: Theory, Methods and Applications, Volume I. Unilateral Analysis and Unilateral Mechanics, Nonconvex Optimization and its Applications, 69. Kluwer Academic Publishers, Boston, MA, 2003.
    [15] D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications. Vol. Ⅱ: Unilateral Problems, Nonconvex Optimization and its Applications, 70. Kluwer Academic Publishers, Boston, MA, 2003.
    [16] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.  doi: 10.1080/02331930801951116.
    [17] J.F. HanS. Migórski and H.D. Zeng, Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response, Nonlinear Anal. Real World Appl., 28 (2016), 229-250.  doi: 10.1016/j.nonrwa.2015.10.004.
    [18] W.M. HanS. Migórski and M. Sofonea, Analysis of a general dynamic history-dependent variational-hemivariational inequality, Nonlinear Anal. Real World Appl., 36 (2017), 69-88.  doi: 10.1016/j.nonrwa.2016.12.007.
    [19] W.M. Han and  M. SofoneaQuasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30. American Mathematical Society, Providence, RI, International Press, omerville, MA, 2002. 
    [20] J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and its Applications, 35. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4757-5233-5.
    [21] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.  doi: 10.1007/BF00932654.
    [22] A. Kulig and S. Migórski, Solvability and continuous dependence results for second order nonlinear inclusion with Volterra-type operator, Nonlinear Analysis, 75 (2012), 4729-4746.  doi: 10.1016/j.na.2012.03.023.
    [23] K. Kuttler and M. Shillor, Dynamic contact with Signorini's condition and slip rate dependent friction, Electron. J. Differ. Equ., 2004 (2004), 21 pp.
    [24] S. Migórski, Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction, Appl. Anal., 84 (2005), 669-699.  doi: 10.1080/00036810500048129.
    [25] S. Migórski and A. Ochal, A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83 (2006), 247-275.  doi: 10.1007/s10659-005-9034-0.
    [26] S. MigórskiA. Ochal and M. Sofonea, History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 12 (2011), 3384-3396.  doi: 10.1016/j.nonrwa.2011.06.002.
    [27] 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.
    [28] S. MigórskiA. Ochal and M. Sofonea, History-dependent variational-hemivariational inequalities in contact mechanics, Nonlinear Anal. Real World Appl., 22 (2015), 604-618.  doi: 10.1016/j.nonrwa.2014.09.021.
    [29] S. MigórskiA. Ochal and M. Sofonea, Evolutionary inclusions and hemivariational inequalities, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 39-64.  doi: 10.1007/978-3-319-14490-0_2.
    [30] S. Migórski and J. Ogorzaly, A class of evolution variational inequalities with memory and its application to viscoelastic frictional contact problems, J. Math. Anal. Appl., 442 (2016), 685-702.  doi: 10.1016/j.jmaa.2016.04.076.
    [31] S. Migórski and J. Ogorzaly, Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics, Z. angew. Math. Phys., 68 (2017), Art. 15, 22 pp. doi: 10.1007/s00033-016-0758-4.
    [32] S. MigórskiM. Sofonea and S.D. Zeng, Well-posedness of history-dependent sweeping processes, SIAM J. Math. Anal., 51 (2019), 1082-1107.  doi: 10.1137/18M1201561.
    [33] D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and its Applications, 29. Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-1-4615-4064-9.
    [34] 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.
    [35] P.D. Panagiotopoulos, Nonconvex energy functions, hemivariational inequalities and substationary principles, Acta Mech., 42 (1983), 111-130.  doi: 10.1007/BF01170410.
    [36] P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Boston, Inc., Boston, MA, 1985. doi: 10.1007/978-1-4612-5152-1.
    [37] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1.
    [38] M. Shillor, M. Sofonea and J.J. Telega, Models and Analysis of Quasistatic Contact, Lect. Notes Phys., 655. Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.
    [39] M. Sofonea and  A. MateiMathematical Models in Contact Mechanics, London Mathematical Society Lecture Notes Series, 398. Cambridge University Press, Cambridge, 2012. 
    [40] M. Sofonea and  S. MigórskiVariational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2018. 
    [41] M. SofoneaS. Migórski and A. Ochal, Two history-dependent contact problems, Advances in Variational and Hemivariational Inequalities, Adv. Mech. Math., Springer, Cham, 33 (2015), 355-380.  doi: 10.1007/978-3-319-14490-0_14.
    [42] M. Sofonea, Y.-B. Xiao and S.D. Zeng, Generalized penalty method for history-dependent variational–hemivariational inequalities, submitted.
    [43] M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of split problems, submitted.
    [44] M. Sofonea and Y.-B. Xiao, Tykhonov well-posedness of a viscoplastic contact problem, Evolution Equations and Control Theory, to appear.
    [45] 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.
    [46] Y.-B. Xiao and M. Sofonea, Fully history-dependent quasivariational inequalities in contact mechanics, Appl. Anal., 95 (2016), 2464-2484.  doi: 10.1080/00036811.2015.1093623.
    [47] E. Zeidler, Nonlinear Functional Analysis and Applications, Ⅱ A/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
  • 加载中

Article Metrics

HTML views(455) PDF downloads(289) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint