Skip to main content
SpringerLink
Log in

Differential History-Dependent Variational-Hemivariational Inequality with Application to a Dynamic Contact Problem

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

This paper is dedicated to the discussion of a new dynamical system involving a history-dependent variational-hemivariational inequality coupled with a non-linear evolution equation. The existence and uniqueness of the solution to this problem are established using the Rothe method and a surjectivity result for a pseudo-monotone perturbation of a maximal operator. Additionally, we derive the regularity solution for such a history-dependent variational-hemivariational inequality. Furthermore, the main results obtained in this study are applied to investigate the unique solvability of a dynamical viscoelastic frictional contact problem with long memory and wear.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, K.T., Shillor, M., Wright, S., Klarbring, A.: A dynamic thermoviscoelastic contact problem with friction and wear. Int. J. Eng. Sci. 35, 1291–1309 (1997)

    Article  MathSciNet  Google Scholar 

  2. Bartosz, K., Sofonea, M.: The Rothe method for variational-hemivariational inequalities with applications to contact mechanics. SIAM J. Math. Anal. 48, 861–883 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bartosz, K., Cheng, X., Kalita, Y.P.Y., Zheng, C.: Rothe method for parabolic variational–hemivariational inequalities. J. Math. Anal. Appl. 423, 841–862 (2015)

    Article  MathSciNet  Google Scholar 

  4. Cai, D.L., Migórski, S., Xiao, Y.B.: Optimal control of differential quasi-variational-hemivariational inequalities with applications. Sci. China Math. 71, 23p (2023)

    Google Scholar 

  5. Carstensen, C., Gwinner, J.: A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. Ann. Mat. Pura Appl. 177, 363–394 (1999)

    Article  MathSciNet  Google Scholar 

  6. Chen, T., Huang, N.J., Xiao, Y.B.: Variational and numerical analysis of a dynamic viscoelastic contact problem with friction and wear. Optimization 69, 2003–2031 (2020)

    Article  MathSciNet  Google Scholar 

  7. Chen, T., Huang, N.J., Li, X.S., Zou, Y.Z.: A new class of differential nonlinear system involving parabolic varitional and history-dependent hemivariational inequalities arising in contact mechanics. Commun. Nonlinear Sci. Numer. Simul. 101, 105886 (2021)

    Article  Google Scholar 

  8. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Springer, Berlin (2003)

    Book  Google Scholar 

  9. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. Am. Math. Soc., Providence (2002)

    Book  Google Scholar 

  10. Han, W., Migórski, S., Sofonea, M.: A class of variational–hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)

    Article  MathSciNet  Google Scholar 

  11. Han, W., Migórski, S., Sofonea, M.: Advances in variational and hemivariational inequalities with applications: theory, numerical analysis, and applications. In: Adv. Mech. Math., vol. 33. Springer, Berlin (2015)

    Google Scholar 

  12. Han, W., Migórski, S., Sofonea, M.: Analysis of a general dynamic history-dependent variational- hemivariational inequality. Nonlinear Anal., Real World Appl. 36, 69–88 (2017)

    Article  MathSciNet  Google Scholar 

  13. Kalita, P.: Regularity and Rothe method error estimates for parabolic hemivariational inequality. J. Math. Anal. Appl. 389(1), 618–631 (2012)

    Article  MathSciNet  Google Scholar 

  14. Migórski, S.: Existence of solutions for a class of history-dependent evolution hemivariational-inequalities. Dyn. Syst. Appl. 21, 319–330 (2012)

    MathSciNet  Google Scholar 

  15. Migórski, S., Cai, D.L.: A new system of differential quasi-hemivariational inequalities in contact mechanics. Appl. Math. Optim. 88, 20 (2023)

    Article  MathSciNet  Google Scholar 

  16. Migórski, S., Cai, D.L.: A general differential quasi variational-hemivariational inequality: well-posedness and application. Commun. Nonlinear Sci. Numer. Simul. 125, 107379, 29p (2023)

    Article  MathSciNet  Google Scholar 

  17. Migórski, S., Ochal, A.: Quasistatic hemivariational inequality via vanishing acceleration approach. SIAM J. Math. Anal. 41, 1415–1435 (2009)

    Article  MathSciNet  Google Scholar 

  18. Migórski, S., Ogorzały, J.: A class of evolution variational inequalities with memory and its application to viscoelastic frictional contact problems. J. Math. Anal. Appl. 442, 685–702 (2016)

    Article  MathSciNet  Google Scholar 

  19. Migórski, S., Ogorzaly, J.: Dynamic history-dependent variational-hemivariational inequalities with applications to contact mechanics. Z. Angew. Math. Phys. 68, 15 (2017)

    Article  MathSciNet  Google Scholar 

  20. Migórski, S., Zeng, S.: Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model. Nonlinear Anal., Real World Appl. 43, 121–143 (2018)

    Article  MathSciNet  Google Scholar 

  21. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models. Anal. Cont. Prob. Adv. Mech. Math., vol. 26. Springer, New York (2013)

    Google Scholar 

  22. Migórski, S., Han, W., Zeng, S.: A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations. Eur. J. Appl. Math. 32(1), 59–88 (2021)

    Article  MathSciNet  Google Scholar 

  23. Migórski, S., Cai, D.L., Dudek, S.: Differential variational-hemivariational inequalities with application to contact mechanics. Nonlinear Anal., Real World Appl. 71, 103816 (2023)

    Article  MathSciNet  Google Scholar 

  24. Sofonea, M., Matei, A.: History-dependent quasi-variational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491 (2011)

    Article  MathSciNet  Google Scholar 

  25. Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Notes, vol. 398. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  26. Sofonea, M., Han, W.M., Shillor, M.: Analysis and Approximation of Contact Problems with Adhesion or Damage. CRC Press, Boca Raton (2005)

    Book  Google Scholar 

  27. Sofonea, M., Patrulescu, F., Souleiman, Y.: Analysis of a contact problem with wear and unilateral constraint. Appl. Anal. 95, 2590–2607 (2016)

    Article  MathSciNet  Google Scholar 

  28. Weng, Y.H., Chen, T., Li, X.S., Huang, N.J.: Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application. Comput. Math. Appl. 98, 0898 (2021)

    Article  MathSciNet  Google Scholar 

  29. Zeidler, E.: Nonlinear Functional Analysis and Applications. Springer, New York (1990)

    Book  Google Scholar 

  30. Zeng, S., Migórski, S., Nguyen, V.T.: A class of hyperbolic variational–hemivariational inequalities without damping terms. Adv. Nonlinear Anal. 11(1), 1287–1306 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Polydisciplinary Faculty of Khouribga, University Sultan Moulay Slimane, Beni Mellal, Morocco

    Abderrahmane Oultou, Zakaria Faiz & Hicham Benaissa

  2. Polydisciplinary Faculty of Ouarzazate, Ibn Zohr University, Agadir, Morocco

    Othmane Baiz

Authors
  1. Abderrahmane Oultou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zakaria Faiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Othmane Baiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hicham Benaissa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zakaria Faiz.

Ethics declarations

Competing Interests

There is no conflict of interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oultou, A., Faiz, Z., Baiz, O. et al. Differential History-Dependent Variational-Hemivariational Inequality with Application to a Dynamic Contact Problem. Acta Appl Math 189, 8 (2024). https://doi.org/10.1007/s10440-024-00637-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-024-00637-2

Keywords

Mathematics Subject Classification

Search

Navigation